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Mathematics · Textbook solutions

Integrals

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 314 questions

  1. 7.1 Eg.1
    Write an anti derivative for each of the following functions using the method of inspection: (i) cos⁡2x\cos 2x (ii) 3x2+4x33x^2 + 4x^3 (iii) 1x\dfrac{1}{x}, x≠0x \neq 0.
  2. 7.1 Eg.2
    Find the following integrals: (i) ∫x3−1x2 dx\int \dfrac{x^3 - 1}{x^2}\,dx (ii) ∫(x2/3+1)dx\int \left(x^{2/3} + 1\right) dx (iii) ∫(x3/2+2ex−1x)dx\int \left(x^{3/2} + 2e^x - \dfrac{1}{x}\right) dx.
  3. 7.1 Eg.3
    Find the following integrals: (i) ∫(sin⁡x+cos⁡x) dx\int (\sin x + \cos x)\,dx (ii) ∫csc⁡x (csc⁡x+cot⁡x) dx\int \csc x\,(\csc x + \cot x)\,dx (iii) ∫1−sin⁡xcos⁡2x dx\int \dfrac{1 - \sin x}{\cos^2 x}\,dx.
  4. 7.1 Eg.4
    Find the anti derivative FF of ff defined by f(x)=4x3−6f(x) = 4x^3 - 6, where F(0)=3F(0) = 3.
  1. Ex 7.1 Q1
    Find an anti derivative (or integral) of the following function by the method of inspection: sin⁡2x\sin 2x.
  2. Ex 7.1 Q2
    Find an anti derivative (or integral) of the following function by the method of inspection: cos⁡3x\cos 3x.
  3. Ex 7.1 Q3
    Find an anti derivative (or integral) of the following function by the method of inspection: e2xe^{2x}.
  4. Ex 7.1 Q4
    Find an anti derivative (or integral) of the following function by the method of inspection: (ax+b)2(ax + b)^2.
  5. Ex 7.1 Q5
    Find an anti derivative (or integral) of the following function by the method of inspection: sin⁡2x−4e3x\sin 2x - 4e^{3x}.
  6. Ex 7.1 Q6
    Find the integral ∫(4e3x+1)dx\int \left(4e^{3x} + 1\right) dx.
  7. Ex 7.1 Q7
    Find the integral ∫x2(1−1x2)dx\int x^2 \left(1 - \dfrac{1}{x^2}\right) dx.
  8. Ex 7.1 Q8
    Find the integral ∫(ax2+bx+c)dx\int \left(ax^2 + bx + c\right) dx.
  9. Ex 7.1 Q9
    Find the integral ∫(2x2+ex)dx\int \left(2x^2 + e^x\right) dx.
  10. Ex 7.1 Q10
    Find the integral ∫(x−1x)2dx\int \left(\sqrt{x} - \dfrac{1}{\sqrt{x}}\right)^2 dx.
  11. Ex 7.1 Q11
    Find the integral ∫x3+5x2−4x2 dx\int \dfrac{x^3 + 5x^2 - 4}{x^2}\,dx.
  12. Ex 7.1 Q12
    Find the integral ∫x3+3x+4x dx\int \dfrac{x^3 + 3x + 4}{\sqrt{x}}\,dx.
  13. Ex 7.1 Q13
    Find the integral ∫x3−x2+x−1x−1 dx\int \dfrac{x^3 - x^2 + x - 1}{x - 1}\,dx.
  14. Ex 7.1 Q14
    Find the integral ∫(1−x)x dx\int (1 - x)\sqrt{x}\,dx.
  15. Ex 7.1 Q15
    Find the integral ∫x(3x2+2x+3)dx\int \sqrt{x}\left(3x^2 + 2x + 3\right) dx.
  16. Ex 7.1 Q16
    Find the integral ∫(2x−3cos⁡x+ex)dx\int \left(2x - 3\cos x + e^x\right) dx.
  17. Ex 7.1 Q17
    Find the integral ∫(2x2−3sin⁡x+5x)dx\int \left(2x^2 - 3\sin x + 5\sqrt{x}\right) dx.
  18. Ex 7.1 Q18
    Find the integral ∫sec⁡x (sec⁡x+tan⁡x) dx\int \sec x\,(\sec x + \tan x)\,dx.
  19. Ex 7.1 Q19
    Find the integral ∫sec⁡2xcsc⁡2x dx\int \dfrac{\sec^2 x}{\csc^2 x}\,dx.
  20. Ex 7.1 Q20
    Find the integral ∫2−3sin⁡xcos⁡2x dx\int \dfrac{2 - 3\sin x}{\cos^2 x}\,dx.
  21. Ex 7.1 Q21
    The anti derivative of (x+1x)\left(\sqrt{x} + \dfrac{1}{\sqrt{x}}\right) equals
    1. A.
      13x1/3+2x1/2+C\dfrac{1}{3}x^{1/3} + 2x^{1/2} + C
    2. B.
      23x2/3+12x2+C\dfrac{2}{3}x^{2/3} + \dfrac{1}{2}x^2 + C
    3. C.
      23x3/2+2x1/2+C\dfrac{2}{3}x^{3/2} + 2x^{1/2} + C
    4. D.
      32x3/2+12x1/2+C\dfrac{3}{2}x^{3/2} + \dfrac{1}{2}x^{1/2} + C
  22. Ex 7.1 Q22
    If ddxf(x)=4x3−3x4\dfrac{d}{dx}f(x) = 4x^3 - \dfrac{3}{x^4} such that f(2)=0f(2) = 0. Then f(x)f(x) is
    1. A.
      x4+1x3−1298x^4 + \dfrac{1}{x^3} - \dfrac{129}{8}
    2. B.
      x3+1x4+1298x^3 + \dfrac{1}{x^4} + \dfrac{129}{8}
    3. C.
      x4+1x3+1298x^4 + \dfrac{1}{x^3} + \dfrac{129}{8}
    4. D.
      x3+1x4−1298x^3 + \dfrac{1}{x^4} - \dfrac{129}{8}
  1. Integrate the following functions w.r.t. xx:
    7.2 Eg.5(i)
    sin⁡mx\sin mx
  2. 7.2 Eg.5(ii)
    2xsin⁡(x2+1)2x \sin (x^2 + 1)
  3. 7.2 Eg.5(iii)
    tan⁡4x sec⁡2xx\dfrac{\tan^4 \sqrt{x}\, \sec^2 \sqrt{x}}{\sqrt{x}}
  4. 7.2 Eg.5(iv)
    sin⁡(tan⁡−1x)1+x2\dfrac{\sin (\tan^{-1} x)}{1 + x^2}
  5. Find the following integrals:
    7.2 Eg.6(i)
    ∫sin⁡3xcos⁡2x dx\int \sin^3 x \cos^2 x\,dx
  6. 7.2 Eg.6(ii)
    ∫sin⁡xsin⁡(x+a) dx\int \dfrac{\sin x}{\sin (x + a)}\,dx
  7. 7.2 Eg.6(iii)
    ∫11+tan⁡x dx\int \dfrac{1}{1 + \tan x}\,dx
  1. Ex 7.2 Q1
    Integrate the function 2x1+x2\dfrac{2x}{1 + x^2}.
  2. Ex 7.2 Q2
    Integrate the function (log⁡x)2x\dfrac{(\log x)^2}{x}.
  3. Ex 7.2 Q3
    Integrate the function 1x+xlog⁡x\dfrac{1}{x + x \log x}.
  4. Ex 7.2 Q4
    Integrate the function sin⁡xsin⁡(cos⁡x)\sin x \sin (\cos x).
  5. Ex 7.2 Q5
    Integrate the function sin⁡(ax+b)cos⁡(ax+b)\sin (ax + b) \cos (ax + b).
  6. Ex 7.2 Q6
    Integrate the function ax+b\sqrt{ax + b}.
  7. Ex 7.2 Q7
    Integrate the function xx+2x \sqrt{x + 2}.
  8. Ex 7.2 Q8
    Integrate the function x1+2x2x \sqrt{1 + 2x^2}.
  9. Ex 7.2 Q9
    Integrate the function (4x+2)x2+x+1(4x + 2) \sqrt{x^2 + x + 1}.
  10. Ex 7.2 Q10
    Integrate the function 1x−x\dfrac{1}{x - \sqrt{x}}.
  11. Ex 7.2 Q11
    Integrate the function xx+4, x>0\dfrac{x}{\sqrt{x + 4}},\ x > 0.
  12. Ex 7.2 Q12
    Integrate the function (x3−1)13x5(x^3 - 1)^{\frac{1}{3}} x^5.
  13. Ex 7.2 Q13
    Integrate the function x2(2+3x3)3\dfrac{x^2}{(2 + 3x^3)^3}.
  14. Ex 7.2 Q14
    Integrate the function 1x(log⁡x)m, x>0, m≠1\dfrac{1}{x (\log x)^m},\ x > 0,\ m \ne 1.
  15. Ex 7.2 Q15
    Integrate the function x9−4x2\dfrac{x}{9 - 4x^2}.
  16. Ex 7.2 Q16
    Integrate the function e2x+3e^{2x + 3}.
  17. Ex 7.2 Q17
    Integrate the function xex2\dfrac{x}{e^{x^2}}.
  18. Ex 7.2 Q18
    Integrate the function etan⁡−1x1+x2\dfrac{e^{\tan^{-1} x}}{1 + x^2}.
  19. Ex 7.2 Q19
    Integrate the function e2x−1e2x+1\dfrac{e^{2x} - 1}{e^{2x} + 1}.
  20. Ex 7.2 Q20
    Integrate the function e2x−e−2xe2x+e−2x\dfrac{e^{2x} - e^{-2x}}{e^{2x} + e^{-2x}}.
  21. Ex 7.2 Q21
    Integrate the function tan⁡2(2x−3)\tan^2 (2x - 3).
  22. Ex 7.2 Q22
    Integrate the function sec⁡2(7−4x)\sec^2 (7 - 4x).
  23. Ex 7.2 Q23
    Integrate the function sin⁡−1x1−x2\dfrac{\sin^{-1} x}{\sqrt{1 - x^2}}.
  24. Ex 7.2 Q24
    Integrate the function 2cos⁡x−3sin⁡x6cos⁡x+4sin⁡x\dfrac{2 \cos x - 3 \sin x}{6 \cos x + 4 \sin x}.
  25. Ex 7.2 Q25
    Integrate the function 1cos⁡2x(1−tan⁡x)2\dfrac{1}{\cos^2 x (1 - \tan x)^2}.
  26. Ex 7.2 Q26
    Integrate the function cos⁡xx\dfrac{\cos \sqrt{x}}{\sqrt{x}}.
  27. Ex 7.2 Q27
    Integrate the function sin⁡2x cos⁡2x\sqrt{\sin 2x}\, \cos 2x.
  28. Ex 7.2 Q28
    Integrate the function cos⁡x1+sin⁡x\dfrac{\cos x}{\sqrt{1 + \sin x}}.
  29. Ex 7.2 Q29
    Integrate the function cot⁡xlog⁡sin⁡x\cot x \log \sin x.
  30. Ex 7.2 Q30
    Integrate the function sin⁡x1+cos⁡x\dfrac{\sin x}{1 + \cos x}.
  31. Ex 7.2 Q31
    Integrate the function sin⁡x(1+cos⁡x)2\dfrac{\sin x}{(1 + \cos x)^2}.
  32. Ex 7.2 Q32
    Integrate the function 11+cot⁡x\dfrac{1}{1 + \cot x}.
  33. Ex 7.2 Q33
    Integrate the function 11−tan⁡x\dfrac{1}{1 - \tan x}.
  34. Ex 7.2 Q34
    Integrate the function tan⁡xsin⁡xcos⁡x\dfrac{\sqrt{\tan x}}{\sin x \cos x}.
  35. Ex 7.2 Q35
    Integrate the function (1+log⁡x)2x\dfrac{(1 + \log x)^2}{x}.
  36. Ex 7.2 Q36
    Integrate the function (x+1)(x+log⁡x)2x\dfrac{(x + 1)(x + \log x)^2}{x}.
  37. Ex 7.2 Q37
    Integrate the function x3sin⁡(tan⁡−1x4)1+x8\dfrac{x^3 \sin (\tan^{-1} x^4)}{1 + x^8}.
  38. Ex 7.2 Q38
    ∫10x9+10xlog⁡e10x10+10x dx\displaystyle\int \dfrac{10x^9 + 10^x \log_e 10}{x^{10} + 10^x}\,dx equals
    1. A.
      10x−x10+C10^x - x^{10} + C
    2. B.
      10x+x10+C10^x + x^{10} + C
    3. C.
      (10x−x10)−1+C(10^x - x^{10})^{-1} + C
    4. D.
      log⁡(10x+x10)+C\log (10^x + x^{10}) + C
  39. Ex 7.2 Q39
    ∫dxsin⁡2xcos⁡2x\displaystyle\int \dfrac{dx}{\sin^2 x \cos^2 x} equals
    1. A.
      tan⁡x+cot⁡x+C\tan x + \cot x + C
    2. B.
      tan⁡x−cot⁡x+C\tan x - \cot x + C
    3. C.
      tan⁡xcot⁡x+C\tan x \cot x + C
    4. D.
      tan⁡x−cot⁡2x+C\tan x - \cot 2x + C
  1. Find the following integrals.
    7.3 Eg.7(i)
    Find ∫cos⁡2x dx\int \cos^2 x\,dx.
  2. 7.3 Eg.7(ii)
    Find ∫sin⁡2xcos⁡3x dx\int \sin 2x \cos 3x\,dx.
  3. 7.3 Eg.7(iii)
    Find ∫sin⁡3x dx\int \sin^3 x\,dx.
  1. Ex 7.3 Q1
    Find the integral of sin⁡2(2x+5)\sin^2 (2x + 5).
  2. Ex 7.3 Q2
    Find the integral of sin⁡3xcos⁡4x\sin 3x \cos 4x.
  3. Ex 7.3 Q3
    Find the integral of cos⁡2xcos⁡4xcos⁡6x\cos 2x \cos 4x \cos 6x.
  4. Ex 7.3 Q4
    Find the integral of sin⁡3(2x+1)\sin^3 (2x + 1).
  5. Ex 7.3 Q5
    Find the integral of sin⁡3xcos⁡3x\sin^3 x \cos^3 x.
  6. Ex 7.3 Q6
    Find the integral of sin⁡xsin⁡2xsin⁡3x\sin x \sin 2x \sin 3x.
  7. Ex 7.3 Q7
    Find the integral of sin⁡4xsin⁡8x\sin 4x \sin 8x.
  8. Ex 7.3 Q8
    Find the integral of 1−cos⁡x1+cos⁡x\dfrac{1 - \cos x}{1 + \cos x}.
  9. Ex 7.3 Q9
    Find the integral of cos⁡x1+cos⁡x\dfrac{\cos x}{1 + \cos x}.
  10. Ex 7.3 Q10
    Find the integral of sin⁡4x\sin^4 x.
  11. Ex 7.3 Q11
    Find the integral of cos⁡42x\cos^4 2x.
  12. Ex 7.3 Q12
    Find the integral of sin⁡2x1+cos⁡x\dfrac{\sin^2 x}{1 + \cos x}.
  13. Ex 7.3 Q13
    Find the integral of cos⁡2x−cos⁡2αcos⁡x−cos⁡α\dfrac{\cos 2x - \cos 2\alpha}{\cos x - \cos \alpha}.
  14. Ex 7.3 Q14
    Find the integral of cos⁡x−sin⁡x1+sin⁡2x\dfrac{\cos x - \sin x}{1 + \sin 2x}.
  15. Ex 7.3 Q15
    Find the integral of tan⁡32xsec⁡2x\tan^3 2x \sec 2x.
  16. Ex 7.3 Q16
    Find the integral of tan⁡4x\tan^4 x.
  17. Ex 7.3 Q17
    Find the integral of sin⁡3x+cos⁡3xsin⁡2xcos⁡2x\dfrac{\sin^3 x + \cos^3 x}{\sin^2 x \cos^2 x}.
  18. Ex 7.3 Q18
    Find the integral of cos⁡2x+2sin⁡2xcos⁡2x\dfrac{\cos 2x + 2\sin^2 x}{\cos^2 x}.
  19. Ex 7.3 Q19
    Find the integral of 1sin⁡xcos⁡3x\dfrac{1}{\sin x \cos^3 x}.
  20. Ex 7.3 Q20
    Find the integral of cos⁡2x(cos⁡x+sin⁡x)2\dfrac{\cos 2x}{(\cos x + \sin x)^2}.
  21. Ex 7.3 Q21
    Find the integral of sin⁡−1(cos⁡x)\sin^{-1} (\cos x).
  22. Ex 7.3 Q22
    Find the integral of 1cos⁡(x−a)cos⁡(x−b)\dfrac{1}{\cos (x - a) \cos (x - b)}.
  23. Ex 7.3 Q23
    ∫sin⁡2x−cos⁡2xsin⁡2xcos⁡2x dx\displaystyle\int \dfrac{\sin^2 x - \cos^2 x}{\sin^2 x \cos^2 x}\,dx is equal to
    1. A.
      tan⁡x+cot⁡x+C\tan x + \cot x + C
    2. B.
      tan⁡x+csc⁡x+C\tan x + \csc x + C
    3. C.
      −tan⁡x+cot⁡x+C-\tan x + \cot x + C
    4. D.
      tan⁡x+sec⁡x+C\tan x + \sec x + C
  24. Ex 7.3 Q24
    ∫ex(1+x)cos⁡2(exx) dx\displaystyle\int \dfrac{e^x (1 + x)}{\cos^2 (e^x x)}\,dx equals
    1. A.
      −cot⁡(exx)+C-\cot (e^x x) + C
    2. B.
      tan⁡(xex)+C\tan (x e^x) + C
    3. C.
      tan⁡(ex)+C\tan (e^x) + C
    4. D.
      cot⁡(ex)+C\cot (e^x) + C
  1. 7.4 Eg.8
    Find the following integrals: (i) ∫dxx2−16\int \frac{dx}{x^2-16} (ii) ∫dx2x−x2\int \frac{dx}{\sqrt{2x-x^2}}
  2. 7.4 Eg.9
    Find the following integrals: (i) ∫dxx2−6x+13\int \frac{dx}{x^2-6x+13} (ii) ∫dx3x2+13x−10\int \frac{dx}{3x^2+13x-10} (iii) ∫dx5x2−2x\int \frac{dx}{\sqrt{5x^2-2x}}
  3. 7.4 Eg.10
    Find the following integrals: (i) ∫x+22x2+6x+5 dx\int \frac{x+2}{2x^2+6x+5}\,dx (ii) ∫x+35−4x−x2 dx\int \frac{x+3}{\sqrt{5-4x-x^2}}\,dx
  1. Ex 7.4 Q1
    Integrate 3x2x6+1\frac{3x^2}{x^6+1}.
  2. Ex 7.4 Q2
    Integrate 11+4x2\frac{1}{\sqrt{1+4x^2}}.
  3. Ex 7.4 Q3
    Integrate 1(2−x)2+1\frac{1}{\sqrt{(2-x)^2+1}}.
  4. Ex 7.4 Q4
    Integrate 19−25x2\frac{1}{\sqrt{9-25x^2}}.
  5. Ex 7.4 Q5
    Integrate 3x1+2x4\frac{3x}{1+2x^4}.
  6. Ex 7.4 Q6
    Integrate x21−x6\frac{x^2}{1-x^6}.
  7. Ex 7.4 Q7
    Integrate x−1x2−1\frac{x-1}{\sqrt{x^2-1}}.
  8. Ex 7.4 Q8
    Integrate x2x6+a6\frac{x^2}{\sqrt{x^6+a^6}}.
  9. Ex 7.4 Q9
    Integrate sec⁡2xtan⁡2x+4\frac{\sec^2 x}{\sqrt{\tan^2 x+4}}.
  10. Ex 7.4 Q10
    Integrate 1x2+2x+2\frac{1}{\sqrt{x^2+2x+2}}.
  11. Ex 7.4 Q11
    Integrate 19x2+6x+5\frac{1}{9x^2+6x+5}.
  12. Ex 7.4 Q12
    Integrate 17−6x−x2\frac{1}{\sqrt{7-6x-x^2}}.
  13. Ex 7.4 Q13
    Integrate 1(x−1)(x−2)\frac{1}{\sqrt{(x-1)(x-2)}}.
  14. Ex 7.4 Q14
    Integrate 18+3x−x2\frac{1}{\sqrt{8+3x-x^2}}.
  15. Ex 7.4 Q15
    Integrate 1(x−a)(x−b)\frac{1}{\sqrt{(x-a)(x-b)}}.
  16. Ex 7.4 Q16
    Integrate 4x+12x2+x−3\frac{4x+1}{\sqrt{2x^2+x-3}}.
  17. Ex 7.4 Q17
    Integrate x+2x2−1\frac{x+2}{\sqrt{x^2-1}}.
  18. Ex 7.4 Q18
    Integrate 5x−21+2x+3x2\frac{5x-2}{1+2x+3x^2}.
  19. Ex 7.4 Q19
    Integrate 6x+7(x−5)(x−4)\frac{6x+7}{\sqrt{(x-5)(x-4)}}.
  20. Ex 7.4 Q20
    Integrate x+24x−x2\frac{x+2}{\sqrt{4x-x^2}}.
  21. Ex 7.4 Q21
    Integrate x+2x2+2x+3\frac{x+2}{\sqrt{x^2+2x+3}}.
  22. Ex 7.4 Q22
    Integrate x+3x2−2x−5\frac{x+3}{x^2-2x-5}.
  23. Ex 7.4 Q23
    Integrate 5x+3x2+4x+10\frac{5x+3}{\sqrt{x^2+4x+10}}.
  24. Ex 7.4 Q24
    ∫dxx2+2x+2\int \frac{dx}{x^2+2x+2} equals
    1. A.
      xtan⁡−1(x+1)+Cx\tan^{-1}(x+1) + C
    2. B.
      tan⁡−1(x+1)+C\tan^{-1}(x+1) + C
    3. C.
      (x+1)tan⁡−1x+C(x+1)\tan^{-1}x + C
    4. D.
      tan⁡−1x+C\tan^{-1}x + C
  25. Ex 7.4 Q25
    ∫dx9x−4x2\int \frac{dx}{\sqrt{9x-4x^2}} equals
    1. A.
      19sin⁡−1(9x−88)+C\frac{1}{9}\sin^{-1}\left(\frac{9x-8}{8}\right) + C
    2. B.
      12sin⁡−1(8x−99)+C\frac{1}{2}\sin^{-1}\left(\frac{8x-9}{9}\right) + C
    3. C.
      13sin⁡−1(9x−88)+C\frac{1}{3}\sin^{-1}\left(\frac{9x-8}{8}\right) + C
    4. D.
      12sin⁡−1(9x−89)+C\frac{1}{2}\sin^{-1}\left(\frac{9x-8}{9}\right) + C
  1. 7.5 Eg.11
    Find ∫dx(x+1)(x+2)\int \frac{dx}{(x+1)(x+2)}.
  2. 7.5 Eg.12
    Find ∫x2+1x2−5x+6 dx\int \frac{x^2+1}{x^2-5x+6}\,dx.
  3. 7.5 Eg.13
    Find ∫3x−2(x+1)2(x+3) dx\int \frac{3x-2}{(x+1)^2(x+3)}\,dx.
  4. 7.5 Eg.14
    Find ∫x2(x2+1)(x2+4) dx\int \frac{x^2}{(x^2+1)(x^2+4)}\,dx.
  5. 7.5 Eg.15
    Find ∫(3sin⁡ϕ−2)cos⁡ϕ5−cos⁡2ϕ−4sin⁡ϕ dϕ\int \frac{(3\sin\phi-2)\cos\phi}{5-\cos^2\phi-4\sin\phi}\,d\phi.
  6. 7.5 Eg.16
    Find ∫x2+x+1(x+2)(x2+1) dx\int \frac{x^2+x+1}{(x+2)(x^2+1)}\,dx.
  1. Ex 7.5 Q1
    Integrate the rational function x(x+1)(x+2)\frac{x}{(x+1)(x+2)}.
  2. Ex 7.5 Q2
    Integrate the rational function 1x2−9\frac{1}{x^2-9}.
  3. Ex 7.5 Q3
    Integrate the rational function 3x−1(x−1)(x−2)(x−3)\frac{3x-1}{(x-1)(x-2)(x-3)}.
  4. Ex 7.5 Q4
    Integrate the rational function x(x−1)(x−2)(x−3)\frac{x}{(x-1)(x-2)(x-3)}.
  5. Ex 7.5 Q5
    Integrate the rational function 2xx2+3x+2\frac{2x}{x^2+3x+2}.
  6. Ex 7.5 Q6
    Integrate the rational function 1−x2x(1−2x)\frac{1-x^2}{x(1-2x)}.
  7. Ex 7.5 Q7
    Integrate the rational function x(x2+1)(x−1)\frac{x}{(x^2+1)(x-1)}.
  8. Ex 7.5 Q8
    Integrate the rational function x(x−1)2(x+2)\frac{x}{(x-1)^2(x+2)}.
  9. Ex 7.5 Q9
    Integrate the rational function 3x+5x3−x2−x+1\frac{3x+5}{x^3-x^2-x+1}.
  10. Ex 7.5 Q10
    Integrate the rational function 2x−3(x2−1)(2x+3)\frac{2x-3}{(x^2-1)(2x+3)}.
  11. Ex 7.5 Q11
    Integrate the rational function 5x(x+1)(x2−4)\frac{5x}{(x+1)(x^2-4)}.
  12. Ex 7.5 Q12
    Integrate the rational function x3+x+1x2−1\frac{x^3+x+1}{x^2-1}.
  13. Ex 7.5 Q13
    Integrate the rational function 2(1−x)(1+x2)\frac{2}{(1-x)(1+x^2)}.
  14. Ex 7.5 Q14
    Integrate the rational function 3x−1(x+2)2\frac{3x-1}{(x+2)^2}.
  15. Ex 7.5 Q15
    Integrate the rational function 1x4−1\frac{1}{x^4-1}.
  16. Ex 7.5 Q16
    Integrate the rational function 1x(xn+1)\frac{1}{x(x^n+1)}. [Hint: multiply numerator and denominator by xn−1x^{n-1} and put xn=tx^n=t]
  17. Ex 7.5 Q17
    Integrate the rational function cos⁡x(1−sin⁡x)(2−sin⁡x)\frac{\cos x}{(1-\sin x)(2-\sin x)}. [Hint: Put sin⁡x=t\sin x=t]
  18. Ex 7.5 Q18
    Integrate the rational function (x2+1)(x2+2)(x2+3)(x2+4)\frac{(x^2+1)(x^2+2)}{(x^2+3)(x^2+4)}.
  19. Ex 7.5 Q19
    Integrate the rational function 2x(x2+1)(x2+3)\frac{2x}{(x^2+1)(x^2+3)}.
  20. Ex 7.5 Q20
    Integrate the rational function 1x(x4−1)\frac{1}{x(x^4-1)}.
  21. Ex 7.5 Q21
    Integrate the rational function 1ex−1\frac{1}{e^x-1}. [Hint: Put ex=te^x=t]
  22. Ex 7.5 Q22
    ∫x dx(x−1)(x−2)\int \frac{x\,dx}{(x-1)(x-2)} equals
    1. A.
      log⁡∣(x−1)2x−2∣+C\log\left|\frac{(x-1)^2}{x-2}\right| + C
    2. B.
      log⁡∣(x−2)2x−1∣+C\log\left|\frac{(x-2)^2}{x-1}\right| + C
    3. C.
      log⁡∣(x−1x−2)2∣+C\log\left|\left(\frac{x-1}{x-2}\right)^2\right| + C
    4. D.
      log⁡∣(x−1)(x−2)∣+C\log|(x-1)(x-2)| + C
  23. Ex 7.5 Q23
    ∫dxx(x2+1)\int \frac{dx}{x(x^2+1)} equals
    1. A.
      log⁡∣x∣−12log⁡(x2+1)+C\log|x| - \frac{1}{2}\log(x^2+1) + C
    2. B.
      log⁡∣x∣+12log⁡(x2+1)+C\log|x| + \frac{1}{2}\log(x^2+1) + C
    3. C.
      −log⁡∣x∣+12log⁡(x2+1)+C-\log|x| + \frac{1}{2}\log(x^2+1) + C
    4. D.
      12log⁡∣x∣+log⁡(x2+1)+C\frac{1}{2}\log|x| + \log(x^2+1) + C
  1. 7.6 Eg.17
    Find ∫xcos⁡x dx\int x \cos x \, dx.
  2. 7.6 Eg.18
    Find ∫log⁡x dx\int \log x \, dx.
  3. 7.6 Eg.19
    Find ∫x ex dx\int x\, e^{x} \, dx.
  4. 7.6 Eg.20
    Find ∫xsin⁡−1x1−x2 dx\int \frac{x \sin^{-1} x}{\sqrt{1-x^{2}}} \, dx.
  5. 7.6 Eg.21
    Find ∫exsin⁡x dx\int e^{x} \sin x \, dx.
  6. Find the following integrals.
    7.6 Eg.22(i)
    Find ∫ex(tan⁡−1x+11+x2)dx\int e^{x}\left(\tan^{-1} x + \frac{1}{1+x^{2}}\right) dx.
  7. 7.6 Eg.22(ii)
    Find ∫(x2+1) ex(x+1)2 dx\int \frac{(x^{2}+1)\, e^{x}}{(x+1)^{2}} \, dx.
  1. Ex 7.6 Q1
    Integrate the function xsin⁡xx \sin x.
  2. Ex 7.6 Q2
    Integrate the function xsin⁡3xx \sin 3x.
  3. Ex 7.6 Q3
    Integrate the function x2exx^{2} e^{x}.
  4. Ex 7.6 Q4
    Integrate the function xlog⁡xx \log x.
  5. Ex 7.6 Q5
    Integrate the function xlog⁡2xx \log 2x.
  6. Ex 7.6 Q6
    Integrate the function x2log⁡xx^{2} \log x.
  7. Ex 7.6 Q7
    Integrate the function xsin⁡−1xx \sin^{-1} x.
  8. Ex 7.6 Q8
    Integrate the function xtan⁡−1xx \tan^{-1} x.
  9. Ex 7.6 Q9
    Integrate the function xcos⁡−1xx \cos^{-1} x.
  10. Ex 7.6 Q10
    Integrate the function (sin⁡−1x)2(\sin^{-1} x)^{2}.
  11. Ex 7.6 Q11
    Integrate the function xcos⁡−1x1−x2\dfrac{x \cos^{-1} x}{\sqrt{1-x^{2}}}.
  12. Ex 7.6 Q12
    Integrate the function xsec⁡2xx \sec^{2} x.
  13. Ex 7.6 Q13
    Integrate the function tan⁡−1x\tan^{-1} x.
  14. Ex 7.6 Q14
    Integrate the function x(log⁡x)2x (\log x)^{2}.
  15. Ex 7.6 Q15
    Integrate the function (x2+1)log⁡x(x^{2}+1) \log x.
  16. Ex 7.6 Q16
    Integrate the function ex(sin⁡x+cos⁡x)e^{x} (\sin x + \cos x).
  17. Ex 7.6 Q17
    Integrate the function x ex(1+x)2\dfrac{x\, e^{x}}{(1+x)^{2}}.
  18. Ex 7.6 Q18
    Integrate the function ex(1+sin⁡x1+cos⁡x)e^{x}\left(\dfrac{1+\sin x}{1+\cos x}\right).
  19. Ex 7.6 Q19
    Integrate the function ex(1x−1x2)e^{x}\left(\dfrac{1}{x} - \dfrac{1}{x^{2}}\right).
  20. Ex 7.6 Q20
    Integrate the function (x−3) ex(x−1)3\dfrac{(x-3)\, e^{x}}{(x-1)^{3}}.
  21. Ex 7.6 Q21
    Integrate the function e2xsin⁡xe^{2x} \sin x.
  22. Ex 7.6 Q22
    Integrate the function sin⁡−1(2x1+x2)\sin^{-1}\left(\dfrac{2x}{1+x^{2}}\right).
  23. Ex 7.6 Q23
    ∫x2ex3 dx\int x^{2} e^{x^{3}} \, dx equals
    1. A.
      13ex3+C\frac{1}{3} e^{x^{3}} + C
    2. B.
      13ex2+C\frac{1}{3} e^{x^{2}} + C
    3. C.
      12ex3+C\frac{1}{2} e^{x^{3}} + C
    4. D.
      12ex2+C\frac{1}{2} e^{x^{2}} + C
  24. Ex 7.6 Q24
    ∫exsec⁡x (1+tan⁡x) dx\int e^{x} \sec x\, (1 + \tan x) \, dx equals
    1. A.
      excos⁡x+Ce^{x} \cos x + C
    2. B.
      exsec⁡x+Ce^{x} \sec x + C
    3. C.
      exsin⁡x+Ce^{x} \sin x + C
    4. D.
      extan⁡x+Ce^{x} \tan x + C
  1. 7.7 Eg.23
    Find ∫x2+2x+5 dx\int \sqrt{x^2 + 2x + 5}\,dx.
  2. 7.7 Eg.24
    Find ∫3−2x−x2 dx\int \sqrt{3 - 2x - x^2}\,dx.
  1. Ex 7.7 Q1
    Integrate the function 4−x2\sqrt{4 - x^2}.
  2. Ex 7.7 Q2
    Integrate the function 1−4x2\sqrt{1 - 4x^2}.
  3. Ex 7.7 Q3
    Integrate the function x2+4x+6\sqrt{x^2 + 4x + 6}.
  4. Ex 7.7 Q4
    Integrate the function x2+4x+1\sqrt{x^2 + 4x + 1}.
  5. Ex 7.7 Q5
    Integrate the function 1−4x−x2\sqrt{1 - 4x - x^2}.
  6. Ex 7.7 Q6
    Integrate the function x2+4x−5\sqrt{x^2 + 4x - 5}.
  7. Ex 7.7 Q7
    Integrate the function 1+3x−x2\sqrt{1 + 3x - x^2}.
  8. Ex 7.7 Q8
    Integrate the function x2+3x\sqrt{x^2 + 3x}.
  9. Ex 7.7 Q9
    Integrate the function 1+x29\sqrt{1 + \dfrac{x^2}{9}}.
  10. Ex 7.7 Q10
    ∫1+x2 dx\int \sqrt{1 + x^2}\,dx is equal to
    1. A.
      x21+x2+12log⁡∣(x+1+x2)∣+C\dfrac{x}{2}\sqrt{1 + x^2} + \dfrac{1}{2}\log\left|\left(x + \sqrt{1 + x^2}\right)\right| + C
    2. B.
      23(1+x2)32+C\dfrac{2}{3}(1 + x^2)^{\frac{3}{2}} + C
    3. C.
      23x(1+x2)32+C\dfrac{2}{3}x(1 + x^2)^{\frac{3}{2}} + C
    4. D.
      x221+x2+12x2log⁡∣x+1+x2∣+C\dfrac{x^2}{2}\sqrt{1 + x^2} + \dfrac{1}{2}x^2\log\left|x + \sqrt{1 + x^2}\right| + C
  11. Ex 7.7 Q11
    ∫x2−8x+7 dx\int \sqrt{x^2 - 8x + 7}\,dx is equal to
    1. A.
      12(x−4)x2−8x+7+9log⁡∣x−4+x2−8x+7∣+C\dfrac{1}{2}(x - 4)\sqrt{x^2 - 8x + 7} + 9\log\left|x - 4 + \sqrt{x^2 - 8x + 7}\right| + C
    2. B.
      12(x+4)x2−8x+7+9log⁡∣x+4+x2−8x+7∣+C\dfrac{1}{2}(x + 4)\sqrt{x^2 - 8x + 7} + 9\log\left|x + 4 + \sqrt{x^2 - 8x + 7}\right| + C
    3. C.
      12(x−4)x2−8x+7−32log⁡∣x−4+x2−8x+7∣+C\dfrac{1}{2}(x - 4)\sqrt{x^2 - 8x + 7} - 3\sqrt{2}\log\left|x - 4 + \sqrt{x^2 - 8x + 7}\right| + C
    4. D.
      12(x−4)x2−8x+7−92log⁡∣x−4+x2−8x+7∣+C\dfrac{1}{2}(x - 4)\sqrt{x^2 - 8x + 7} - \dfrac{9}{2}\log\left|x - 4 + \sqrt{x^2 - 8x + 7}\right| + C
  1. Evaluate the following integrals:
    7.8 Eg.25(i)
    Evaluate ∫23x2 dx\int_{2}^{3} x^{2}\,dx.
  2. 7.8 Eg.25(ii)
    Evaluate ∫49x(30−x32)2 dx\int_{4}^{9} \frac{\sqrt{x}}{\left(30 - x^{\frac{3}{2}}\right)^{2}}\,dx.
  3. 7.8 Eg.25(iii)
    Evaluate ∫12x dx(x+1)(x+2)\int_{1}^{2} \frac{x\,dx}{(x+1)(x+2)}.
  4. 7.8 Eg.25(iv)
    Evaluate ∫0π4sin⁡32tcos⁡2t dt\int_{0}^{\frac{\pi}{4}} \sin^{3} 2t \cos 2t \,dt.
  1. Ex 7.8 Q1
    Evaluate the definite integral ∫−11(x+1) dx\int_{-1}^{1} (x+1)\,dx.
  2. Ex 7.8 Q2
    Evaluate the definite integral ∫231x dx\int_{2}^{3} \frac{1}{x}\,dx.
  3. Ex 7.8 Q3
    Evaluate the definite integral ∫12(4x3−5x2+6x+9) dx\int_{1}^{2} (4x^{3} - 5x^{2} + 6x + 9)\,dx.
  4. Ex 7.8 Q4
    Evaluate the definite integral ∫0π4sin⁡2x dx\int_{0}^{\frac{\pi}{4}} \sin 2x \,dx.
  5. Ex 7.8 Q5
    Evaluate the definite integral ∫0π2cos⁡2x dx\int_{0}^{\frac{\pi}{2}} \cos 2x \,dx.
  6. Ex 7.8 Q6
    Evaluate the definite integral ∫45ex dx\int_{4}^{5} e^{x}\,dx.
  7. Ex 7.8 Q7
    Evaluate the definite integral ∫0π4tan⁡x dx\int_{0}^{\frac{\pi}{4}} \tan x \,dx.
  8. Ex 7.8 Q8
    Evaluate the definite integral ∫π6π4cosec⁡x dx\int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \operatorname{cosec} x \,dx.
  9. Ex 7.8 Q9
    Evaluate the definite integral ∫01dx1−x2\int_{0}^{1} \frac{dx}{\sqrt{1 - x^{2}}}.
  10. Ex 7.8 Q10
    Evaluate the definite integral ∫01dx1+x2\int_{0}^{1} \frac{dx}{1 + x^{2}}.
  11. Ex 7.8 Q11
    Evaluate the definite integral ∫23dxx2−1\int_{2}^{3} \frac{dx}{x^{2} - 1}.
  12. Ex 7.8 Q12
    Evaluate the definite integral ∫0π2cos⁡2x dx\int_{0}^{\frac{\pi}{2}} \cos^{2} x \,dx.
  13. Ex 7.8 Q13
    Evaluate the definite integral ∫23x dxx2+1\int_{2}^{3} \frac{x\,dx}{x^{2} + 1}.
  14. Ex 7.8 Q14
    Evaluate the definite integral ∫012x+35x2+1 dx\int_{0}^{1} \frac{2x + 3}{5x^{2} + 1}\,dx.
  15. Ex 7.8 Q15
    Evaluate the definite integral ∫01x ex2 dx\int_{0}^{1} x\, e^{x^{2}}\,dx.
  16. Ex 7.8 Q16
    Evaluate the definite integral ∫125x2x2+4x+3 dx\int_{1}^{2} \frac{5x^{2}}{x^{2} + 4x + 3}\,dx.
  17. Ex 7.8 Q17
    Evaluate the definite integral ∫0π4(2sec⁡2x+x3+2) dx\int_{0}^{\frac{\pi}{4}} (2\sec^{2} x + x^{3} + 2)\,dx.
  18. Ex 7.8 Q18
    Evaluate the definite integral ∫0π(sin⁡2x2−cos⁡2x2) dx\int_{0}^{\pi} \left(\sin^{2}\frac{x}{2} - \cos^{2}\frac{x}{2}\right)\,dx.
  19. Ex 7.8 Q19
    Evaluate the definite integral ∫026x+3x2+4 dx\int_{0}^{2} \frac{6x + 3}{x^{2} + 4}\,dx.
  20. Ex 7.8 Q20
    Evaluate the definite integral ∫01(x ex+sin⁡πx4) dx\int_{0}^{1} \left(x\, e^{x} + \sin\frac{\pi x}{4}\right)\,dx.
  21. Ex 7.8 Q21
    ∫13dx1+x2\int_{1}^{\sqrt{3}} \frac{dx}{1 + x^{2}} equals
    1. A.
      π3\frac{\pi}{3}
    2. B.
      2π3\frac{2\pi}{3}
    3. C.
      π6\frac{\pi}{6}
    4. D.
      π12\frac{\pi}{12}
  22. Ex 7.8 Q22
    ∫023dx4+9x2\int_{0}^{\frac{2}{3}} \frac{dx}{4 + 9x^{2}} equals
    1. A.
      π6\frac{\pi}{6}
    2. B.
      π12\frac{\pi}{12}
    3. C.
      π24\frac{\pi}{24}
    4. D.
      π4\frac{\pi}{4}
  1. 7.9 Eg.26
    Evaluate ∫−115x4x5+1 dx\int_{-1}^{1} 5x^{4}\sqrt{x^{5}+1}\,dx.
  2. 7.9 Eg.27
    Evaluate ∫01tan⁡−1x1+x2 dx\int_{0}^{1} \frac{\tan^{-1} x}{1+x^{2}}\,dx.
  1. Ex 7.9 Q1
    Evaluate the integral using substitution: ∫01xx2+1 dx\int_{0}^{1} \frac{x}{x^{2}+1}\,dx.
  2. Ex 7.9 Q2
    Evaluate the integral using substitution: ∫0π/2sin⁡ϕ cos⁡5ϕ dϕ\int_{0}^{\pi/2} \sqrt{\sin\phi}\,\cos^{5}\phi\,d\phi.
  3. Ex 7.9 Q3
    Evaluate the integral using substitution: ∫01sin⁡−1(2x1+x2)dx\int_{0}^{1} \sin^{-1}\left(\frac{2x}{1+x^{2}}\right)dx.
  4. Ex 7.9 Q4
    Evaluate the integral using substitution: ∫02xx+2 dx\int_{0}^{2} x\sqrt{x+2}\,dx (Put x+2=t2x+2 = t^{2}).
  5. Ex 7.9 Q5
    Evaluate the integral using substitution: ∫0π/2sin⁡x1+cos⁡2x dx\int_{0}^{\pi/2} \frac{\sin x}{1+\cos^{2} x}\,dx.
  6. Ex 7.9 Q6
    Evaluate the integral using substitution: ∫02dxx+4−x2\int_{0}^{2} \frac{dx}{x+4-x^{2}}.
  7. Ex 7.9 Q7
    Evaluate the integral using substitution: ∫−11dxx2+2x+5\int_{-1}^{1} \frac{dx}{x^{2}+2x+5}.
  8. Ex 7.9 Q8
    Evaluate the integral using substitution: ∫12(1x−12x2)e2x dx\int_{1}^{2} \left(\frac{1}{x} - \frac{1}{2x^{2}}\right) e^{2x}\,dx.
  9. Ex 7.9 Q9
    The value of the integral ∫1/31(x−x3)1/3x4 dx\int_{1/3}^{1} \frac{(x-x^{3})^{1/3}}{x^{4}}\,dx is
    1. A.
      6
    2. B.
      0
    3. C.
      3
    4. D.
      4
  10. Ex 7.9 Q10
    If f(x)=∫0xtsin⁡t dtf(x) = \int_{0}^{x} t\sin t\,dt, then f′(x)f'(x) is
    1. A.
      cos⁡x+xsin⁡x\cos x + x\sin x
    2. B.
      xsin⁡xx\sin x
    3. C.
      xcos⁡xx\cos x
    4. D.
      sin⁡x+xcos⁡x\sin x + x\cos x
  1. 7.10 Eg.28
    Evaluate ∫−12∣x3−x∣ dx\int_{-1}^{2} |x^3 - x|\,dx.
  2. 7.10 Eg.29
    Evaluate ∫−π/4π/4sin⁡2x dx\int_{-\pi/4}^{\pi/4} \sin^2 x\,dx.
  3. 7.10 Eg.30
    Evaluate ∫0πxsin⁡x1+cos⁡2x dx\int_{0}^{\pi} \frac{x \sin x}{1 + \cos^2 x}\,dx.
  4. 7.10 Eg.31
    Evaluate ∫−11sin⁡5xcos⁡4x dx\int_{-1}^{1} \sin^5 x \cos^4 x\,dx.
  5. 7.10 Eg.32
    Evaluate ∫0π/2sin⁡4xsin⁡4x+cos⁡4x dx\int_{0}^{\pi/2} \frac{\sin^4 x}{\sin^4 x + \cos^4 x}\,dx.
  6. 7.10 Eg.33
    Evaluate ∫π/6π/3dx1+tan⁡x\int_{\pi/6}^{\pi/3} \frac{dx}{1 + \sqrt{\tan x}}.
  7. 7.10 Eg.34
    Evaluate ∫0π/2log⁡sin⁡x dx\int_{0}^{\pi/2} \log \sin x\,dx.
  1. Ex 7.10 Q1
    By using the properties of definite integrals, evaluate the integral ∫0π/2cos⁡2x dx\int_{0}^{\pi/2} \cos^2 x\,dx.
  2. Ex 7.10 Q2
    By using the properties of definite integrals, evaluate the integral ∫0π/2sin⁡xsin⁡x+cos⁡x dx\int_{0}^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}}\,dx.
  3. Ex 7.10 Q3
    By using the properties of definite integrals, evaluate the integral ∫0π/2sin⁡3/2xsin⁡3/2x+cos⁡3/2x dx\int_{0}^{\pi/2} \frac{\sin^{3/2} x}{\sin^{3/2} x + \cos^{3/2} x}\,dx.
  4. Ex 7.10 Q4
    By using the properties of definite integrals, evaluate the integral ∫0π/2cos⁡5xsin⁡5x+cos⁡5x dx\int_{0}^{\pi/2} \frac{\cos^5 x}{\sin^5 x + \cos^5 x}\,dx.
  5. Ex 7.10 Q5
    By using the properties of definite integrals, evaluate the integral ∫−55∣x+2∣ dx\int_{-5}^{5} |x + 2|\,dx.
  6. Ex 7.10 Q6
    By using the properties of definite integrals, evaluate the integral ∫28∣x−5∣ dx\int_{2}^{8} |x - 5|\,dx.
  7. Ex 7.10 Q7
    By using the properties of definite integrals, evaluate the integral ∫01x(1−x)n dx\int_{0}^{1} x(1 - x)^n\,dx.
  8. Ex 7.10 Q8
    By using the properties of definite integrals, evaluate the integral ∫0π/4log⁡(1+tan⁡x) dx\int_{0}^{\pi/4} \log(1 + \tan x)\,dx.
  9. Ex 7.10 Q9
    By using the properties of definite integrals, evaluate the integral ∫02x2−x dx\int_{0}^{2} x\sqrt{2 - x}\,dx.
  10. Ex 7.10 Q10
    By using the properties of definite integrals, evaluate the integral ∫0π/2(2log⁡sin⁡x−log⁡sin⁡2x) dx\int_{0}^{\pi/2}(2\log \sin x - \log \sin 2x)\,dx.
  11. Ex 7.10 Q11
    By using the properties of definite integrals, evaluate the integral ∫−π/2π/2sin⁡2x dx\int_{-\pi/2}^{\pi/2} \sin^2 x\,dx.
  12. Ex 7.10 Q12
    By using the properties of definite integrals, evaluate the integral ∫0πx dx1+sin⁡x\int_{0}^{\pi} \frac{x\,dx}{1 + \sin x}.
  13. Ex 7.10 Q13
    By using the properties of definite integrals, evaluate the integral ∫−π/2π/2sin⁡7x dx\int_{-\pi/2}^{\pi/2} \sin^7 x\,dx.
  14. Ex 7.10 Q14
    By using the properties of definite integrals, evaluate the integral ∫02πcos⁡5x dx\int_{0}^{2\pi} \cos^5 x\,dx.
  15. Ex 7.10 Q15
    By using the properties of definite integrals, evaluate the integral ∫0π/2sin⁡x−cos⁡x1+sin⁡xcos⁡x dx\int_{0}^{\pi/2} \frac{\sin x - \cos x}{1 + \sin x \cos x}\,dx.
  16. Ex 7.10 Q16
    By using the properties of definite integrals, evaluate the integral ∫0πlog⁡(1+cos⁡x) dx\int_{0}^{\pi} \log(1 + \cos x)\,dx.
  17. Ex 7.10 Q17
    By using the properties of definite integrals, evaluate the integral ∫0axx+a−x dx\int_{0}^{a} \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a - x}}\,dx.
  18. Ex 7.10 Q18
    By using the properties of definite integrals, evaluate the integral ∫04∣x−1∣ dx\int_{0}^{4} |x - 1|\,dx.
  19. Ex 7.10 Q19
    Show that ∫0af(x)g(x) dx=2∫0af(x) dx\int_{0}^{a} f(x)g(x)\,dx = 2\int_{0}^{a} f(x)\,dx, if ff and gg are defined as f(x)=f(a−x)f(x) = f(a - x) and g(x)+g(a−x)=4g(x) + g(a - x) = 4.
  20. Ex 7.10 Q20
    The value of ∫−π/2π/2(x3+xcos⁡x+tan⁡5x+1) dx\int_{-\pi/2}^{\pi/2}(x^3 + x \cos x + \tan^5 x + 1)\,dx is
    1. A.
      00
    2. B.
      22
    3. C.
      π\pi
    4. D.
      11
  21. Ex 7.10 Q21
    The value of ∫0π/2log⁡(4+3sin⁡x4+3cos⁡x) dx\int_{0}^{\pi/2} \log\left(\frac{4 + 3\sin x}{4 + 3\cos x}\right)\,dx is
    1. A.
      22
    2. B.
      34\frac{3}{4}
    3. C.
      00
    4. D.
      −2-2
  1. Misc Eg.35
    Find ∫cos⁡6x 1+sin⁡6x  dx\int \cos 6x\, \sqrt{1+\sin 6x}\; dx.
  2. Misc Eg.36
    Find ∫(x4−x)14x5 dx\int \frac{(x^4 - x)^{\frac{1}{4}}}{x^5}\, dx.
  3. Misc Eg.37
    Find ∫x4 dx(x−1)(x2+1)\int \frac{x^4\, dx}{(x-1)(x^2+1)}.
  4. Misc Eg.38
    Find ∫[log⁡(log⁡x)+1(log⁡x)2]dx\int \left[\log(\log x) + \frac{1}{(\log x)^2}\right] dx.
  5. Misc Eg.39
    Find ∫[cot⁡x+tan⁡x]dx\int \left[\sqrt{\cot x} + \sqrt{\tan x}\right] dx.
  6. Misc Eg.40
    Find ∫sin⁡2xcos⁡2x dx9−cos⁡4(2x)\int \frac{\sin 2x \cos 2x\, dx}{\sqrt{9 - \cos^4(2x)}}.
  7. Misc Eg.41
    Evaluate ∫−132∣xsin⁡(πx)∣ dx\int_{-1}^{\frac{3}{2}} |x \sin(\pi x)|\, dx.
  8. Misc Eg.42
    Evaluate ∫0πx dxa2cos⁡2x+b2sin⁡2x\int_{0}^{\pi} \frac{x\, dx}{a^2\cos^2 x + b^2\sin^2 x}.
  1. Misc Q1
    Integrate the function 1x−x3\frac{1}{x - x^3}.
  2. Misc Q2
    Integrate the function 1x+a+x+b\frac{1}{\sqrt{x+a} + \sqrt{x+b}}.
  3. Misc Q3
    Integrate the function 1xax−x2\frac{1}{x\sqrt{ax - x^2}}. [Hint: Put x=atx = \frac{a}{t}]
  4. Misc Q4
    Integrate the function 1x2(x4+1)34\frac{1}{x^2 (x^4 + 1)^{\frac{3}{4}}}.
  5. Misc Q5
    Integrate the function 1x12+x13\frac{1}{x^{\frac{1}{2}} + x^{\frac{1}{3}}}. [Hint: 1x12+x13=1x13(1+x16)\frac{1}{x^{\frac{1}{2}} + x^{\frac{1}{3}}} = \frac{1}{x^{\frac{1}{3}}\left(1 + x^{\frac{1}{6}}\right)}, put x=t6x = t^6]
  6. Misc Q6
    Integrate the function 5x(x+1)(x2+9)\frac{5x}{(x+1)(x^2+9)}.
  7. Misc Q7
    Integrate the function sin⁡xsin⁡(x−a)\frac{\sin x}{\sin(x - a)}.
  8. Misc Q8
    Integrate the function e5log⁡x−e4log⁡xe3log⁡x−e2log⁡x\frac{e^{5\log x} - e^{4\log x}}{e^{3\log x} - e^{2\log x}}.
  9. Misc Q9
    Integrate the function cos⁡x4−sin⁡2x\frac{\cos x}{\sqrt{4 - \sin^2 x}}.
  10. Misc Q10
    Integrate the function sin⁡8x−cos⁡8x1−2sin⁡2xcos⁡2x\frac{\sin^8 x - \cos^8 x}{1 - 2\sin^2 x \cos^2 x}.
  11. Misc Q11
    Integrate the function 1cos⁡(x+a)cos⁡(x+b)\frac{1}{\cos(x + a)\cos(x + b)}.
  12. Misc Q12
    Integrate the function x31−x8\frac{x^3}{\sqrt{1 - x^8}}.
  13. Misc Q13
    Integrate the function ex(1+ex)(2+ex)\frac{e^x}{(1 + e^x)(2 + e^x)}.
  14. Misc Q14
    Integrate the function 1(x2+1)(x2+4)\frac{1}{(x^2 + 1)(x^2 + 4)}.
  15. Misc Q15
    Integrate the function cos⁡3x  elog⁡sin⁡x\cos^3 x\; e^{\log \sin x}.
  16. Misc Q16
    Integrate the function e3log⁡x(x4+1)−1e^{3\log x}(x^4 + 1)^{-1}.
  17. Misc Q17
    Integrate the function f′(ax+b) [f(ax+b)]nf'(ax + b)\,[f(ax + b)]^n.
  18. Misc Q18
    Integrate the function 1sin⁡3x sin⁡(x+α)\frac{1}{\sqrt{\sin^3 x \, \sin(x + \alpha)}}.
  19. Misc Q19
    Integrate the function 1−x1+x\sqrt{\frac{1 - \sqrt{x}}{1 + \sqrt{x}}}.
  20. Misc Q20
    Integrate the function 2+sin⁡2x1+cos⁡2x ex\frac{2 + \sin 2x}{1 + \cos 2x}\, e^x.
  21. Misc Q21
    Integrate the function x2+x+1(x+1)2(x+2)\frac{x^2 + x + 1}{(x + 1)^2 (x + 2)}.
  22. Misc Q22
    Integrate the function tan⁡−11−x1+x\tan^{-1}\sqrt{\frac{1 - x}{1 + x}}.
  23. Misc Q23
    Integrate the function x2+1 [log⁡(x2+1)−2log⁡x]x4\frac{\sqrt{x^2 + 1}\,\left[\log(x^2 + 1) - 2\log x\right]}{x^4}.
  24. Misc Q24
    Evaluate the definite integral ∫π2πex(1−sin⁡x1−cos⁡x)dx\int_{\frac{\pi}{2}}^{\pi} e^x\left(\frac{1 - \sin x}{1 - \cos x}\right) dx.
  25. Misc Q25
    Evaluate the definite integral ∫0π4sin⁡xcos⁡xcos⁡4x+sin⁡4x dx\int_{0}^{\frac{\pi}{4}} \frac{\sin x \cos x}{\cos^4 x + \sin^4 x}\, dx.
  26. Misc Q26
    Evaluate the definite integral ∫0π2cos⁡2x dxcos⁡2x+4sin⁡2x\int_{0}^{\frac{\pi}{2}} \frac{\cos^2 x\, dx}{\cos^2 x + 4\sin^2 x}.
  27. Misc Q27
    Evaluate the definite integral ∫π6π3sin⁡x+cos⁡xsin⁡2x dx\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin x + \cos x}{\sqrt{\sin 2x}}\, dx.
  28. Misc Q28
    Evaluate the definite integral ∫01dx1+x−x\int_{0}^{1} \frac{dx}{\sqrt{1 + x} - \sqrt{x}}.
  29. Misc Q29
    Evaluate the definite integral ∫0π4sin⁡x+cos⁡x9+16sin⁡2x dx\int_{0}^{\frac{\pi}{4}} \frac{\sin x + \cos x}{9 + 16\sin 2x}\, dx.
  30. Misc Q30
    Evaluate the definite integral ∫0π2sin⁡2x tan⁡−1(sin⁡x) dx\int_{0}^{\frac{\pi}{2}} \sin 2x \, \tan^{-1}(\sin x)\, dx.
  31. Misc Q31
    Evaluate the definite integral ∫14[∣x−1∣+∣x−2∣+∣x−3∣]dx\int_{1}^{4} \left[|x - 1| + |x - 2| + |x - 3|\right] dx.
  32. Misc Q32
    Prove that ∫13dxx2(x+1)=23+log⁡23\int_{1}^{3} \frac{dx}{x^2 (x + 1)} = \frac{2}{3} + \log \frac{2}{3}.
  33. Misc Q33
    Prove that ∫01x ex dx=1\int_{0}^{1} x\, e^x\, dx = 1.
  34. Misc Q34
    Prove that ∫−11x17cos⁡4x dx=0\int_{-1}^{1} x^{17} \cos^4 x\, dx = 0.
  35. Misc Q35
    Prove that ∫0π2sin⁡3x dx=23\int_{0}^{\frac{\pi}{2}} \sin^3 x\, dx = \frac{2}{3}.
  36. Misc Q36
    Prove that ∫0π42tan⁡3x dx=1−log⁡2\int_{0}^{\frac{\pi}{4}} 2\tan^3 x\, dx = 1 - \log 2.
  37. Misc Q37
    Prove that ∫01sin⁡−1x dx=π2−1\int_{0}^{1} \sin^{-1} x\, dx = \frac{\pi}{2} - 1.
  38. Misc Q38
    ∫dxex+e−x\int \frac{dx}{e^x + e^{-x}} is equal to
    1. A.
      tan⁡−1(ex)+C\tan^{-1}(e^x) + C
    2. B.
      tan⁡−1(e−x)+C\tan^{-1}(e^{-x}) + C
    3. C.
      log⁡(ex−e−x)+C\log(e^x - e^{-x}) + C
    4. D.
      log⁡(ex+e−x)+C\log(e^x + e^{-x}) + C
  39. Misc Q39
    ∫cos⁡2x(sin⁡x+cos⁡x)2 dx\int \frac{\cos 2x}{(\sin x + \cos x)^2}\, dx is equal to
    1. A.
      −1sin⁡x+cos⁡x+C\frac{-1}{\sin x + \cos x} + C
    2. B.
      log⁡∣sin⁡x+cos⁡x∣+C\log|\sin x + \cos x| + C
    3. C.
      log⁡∣sin⁡x−cos⁡x∣+C\log|\sin x - \cos x| + C
    4. D.
      1(sin⁡x+cos⁡x)2\frac{1}{(\sin x + \cos x)^2}
  40. Misc Q40
    If f(a+b−x)=f(x)f(a + b - x) = f(x), then ∫abx f(x) dx\int_{a}^{b} x\, f(x)\, dx is equal to
    1. A.
      a+b2∫abf(b−x) dx\frac{a + b}{2}\int_{a}^{b} f(b - x)\, dx
    2. B.
      a+b2∫abf(b+x) dx\frac{a + b}{2}\int_{a}^{b} f(b + x)\, dx
    3. C.
      b−a2∫abf(x) dx\frac{b - a}{2}\int_{a}^{b} f(x)\, dx
    4. D.
      a+b2∫abf(x) dx\frac{a + b}{2}\int_{a}^{b} f(x)\, dx