Indefinite Integration

3.1 SolvedEx.1 $\int (x^3 + 3^x)\, dx$
3.1 SolvedEx.2 $\int \left( \sin x + \dfrac{1}{x} + \dfrac{1}{\sqrt[3]{x}} \right) dx$
3.1 SolvedEx.3 $\int (\tan x + \cot x)^2\, dx$
3.1 SolvedEx.4 $\int \dfrac{\sqrt{x} + 1}{x + \sqrt{x}}\, dx$
3.1 SolvedEx.5 $\int \dfrac{e^{4 \log x} - e^{5 \log x}}{x^5}\, dx$
3.1 SolvedEx.6 $\int \dfrac{2x + 3}{5x - 1}\, dx$
3.1 SolvedEx.7 $\int \dfrac{1}{\sqrt{3x + 1} - \sqrt{3x - 5}}\, dx$
3.1 SolvedEx.8 $\int \dfrac{2x - 7}{\sqrt{3x - 2}}\, dx$
3.1 SolvedEx.9 $\int \dfrac{x^3}{x - 1}\, dx$
3.1 SolvedEx.10 $\int \dfrac{3^x - 4^x}{5^x}\, dx$
3.1 SolvedEx.11 $\int \cos^3 x\, dx$
3.1 SolvedEx.12 $\int \sqrt{1 + \sin 3x}\, dx$
3.1 SolvedEx.13 $\int \sin^4 x\, dx$
3.1 SolvedEx.14 $\int \sin 5x \cdot \cos 7x\, dx$
3.1 SolvedEx.15 $\int \dfrac{\sin^3 x - \cos^3 x}{\sin^2 x \cdot \cos^2 x}\, dx$
3.1 SolvedEx.16 $\int \dfrac{1}{1 - \sin x}\, dx$
3.1 SolvedEx.17 $\int \left( \dfrac{\cos x}{1 - \cos x} \right) dx$
3.1 SolvedEx.18 $\int \dfrac{\cos x - \cos 2x}{1 - \cos x}\, dx$
3.1 SolvedEx.19 $\int \sin^{-1}(\cos 3x)\, dx$
3.1 SolvedEx.20 $\int \tan^{-1}\left( \dfrac{\sin 2x}{1 + \cos 2x} \right) dx$
3.1 SolvedEx.21 $\int \tan^{-1} \sqrt{\dfrac{1 - \sin x}{1 + \sin x}}\, dx$

Integrate the following functions w. r. t. x :
Ex 3.1 I (i) $\int (x^3 + x^2 - x + 1)\, dx$
Ex 3.1 I (ii) $\int x^2 \left(1 - \dfrac{2}{x}\right)^2 dx$
Ex 3.1 I (iii) $\int \left(3 \sec^2 x - \dfrac{4}{x} + \dfrac{1}{x\sqrt{x}} - 7\right) dx$
Ex 3.1 I (iv) $\int \left(2x^3 - 5x + \dfrac{3}{x} + \dfrac{4}{x^5}\right) dx$
Ex 3.1 I (v) $\int \dfrac{3x^3 - 2x + 5}{x\sqrt{x}}\, dx$
Evaluate :
Ex 3.1 II (i) $\int \tan^2 x\, dx$
Ex 3.1 II (ii) $\int \dfrac{\sin 2x}{\cos x}\, dx$
Ex 3.1 II (iii) $\int \dfrac{\sin x}{\cos^2 x}\, dx$
Ex 3.1 II (iv) $\int \dfrac{\cos 2x}{\sin^2 x}\, dx$
Ex 3.1 II (v) $\int \dfrac{\cos 2x}{\sin^2 x \cdot \cos^2 x}\, dx$
Ex 3.1 II (vi) $\int \dfrac{\sin x}{1 + \sin x}\, dx$
Ex 3.1 II (vii) $\int \dfrac{\tan x}{\sec x + \tan x}\, dx$
Ex 3.1 II (viii) $\int \sqrt{1 + \sin 2x}\, dx$
Ex 3.1 II (ix) $\int \sqrt{1 - \cos 2x}\, dx$
Ex 3.1 II (x) $\int \sin 4x \cdot \cos 3x\, dx$
Evaluate :
Ex 3.1 III (i) $\int \dfrac{x}{x + 2}\, dx$
Ex 3.1 III (ii) $\int \dfrac{4x + 3}{2x + 1}\, dx$
Ex 3.1 III (iii) $\int \dfrac{5x + 2}{3x - 4}\, dx$
Ex 3.1 III (iv) $\int \dfrac{x - 2}{\sqrt{x + 5}}\, dx$
Ex 3.1 III (v) $\int \dfrac{2x - 7}{\sqrt{4x - 1}}\, dx$
Ex 3.1 III (vi) $\int \dfrac{\sin 4x}{\cos 2x}\, dx$
Ex 3.1 III (vii) $\int \sqrt{1 + \sin 5x}\, dx$
Ex 3.1 III (viii) $\int \cos^2 x\, dx$
Ex 3.1 III (ix) $\int \dfrac{2}{\sqrt{x} - \sqrt{x + 3}}\, dx$
Ex 3.1 III (x) $\int \dfrac{3}{\sqrt{7x - 2} - \sqrt{7x - 5}}\, dx$
Ex 3.1 IV $f'(x) = x - \dfrac{3}{x^3}$

3.2A SolvedEx.1 $\int \frac{\cot (\log x)}{x}\, dx$
3.2A SolvedEx.2 $\int \frac{\cos \sqrt{x}}{\sqrt{x}}\, dx$
3.2A SolvedEx.3 $\int \frac{\sec^8 x}{\operatorname{cosec} x}\, dx$
3.2A SolvedEx.4 $\int \frac{1}{x + \sqrt{x}}\, dx$
3.2A SolvedEx.5 $\int 5^{5^x} \cdot 5^x\, dx$
3.2A SolvedEx.6 $\int \frac{1}{1 + e^{-x}}\, dx$
3.2A SolvedEx.7 $\int \frac{e^x (1 + x)}{\cos (x \cdot e^x)}\, dx$
3.2A SolvedEx.8 $\int \frac{1}{3x + 7x^{-n}}\, dx$
3.2A SolvedEx.9 $\int (3x + 2)\, \sqrt{x - 4}\, dx$
3.2A SolvedEx.10 $\int \frac{\sin (x + a)}{\cos (x - b)}\, dx$
3.2A SolvedEx.11 $\int \frac{e^x + 1}{e^x - 1}\, dx$
3.2A SolvedEx.12 $\int \frac{1}{1 - \tan x}\, dx$

Integrate the following functions w. r. t. x :
Ex 3.2(A) I (1) $\int \frac{(\log x)^n}{x}\, dx$
Ex 3.2(A) I (2) $\int \frac{(\sin^{-1} x)^{\frac{3}{2}}}{\sqrt{1 - x^2}}\, dx$
Ex 3.2(A) I (3) $\int \frac{1 + x}{x \cdot \sin (x + \log x)}\, dx$
Ex 3.2(A) I (4) $\int \frac{x \cdot \sec^2 (x^2)}{\sqrt{\tan^3 (x^2)}}\, dx$
Ex 3.2(A) I (5) $\int \frac{e^{3x}}{e^{3x} + 1}\, dx$
Ex 3.2(A) I (6) $\int \frac{(x^2 + 2)}{(x^2 + 1)} \cdot a^{x + \tan^{-1} x}\, dx$
Ex 3.2(A) I (7) $\int \frac{e^x \cdot \log (\sin e^x)}{\tan (e^x)}\, dx$
Ex 3.2(A) I (8) $\int \frac{e^{2x} + 1}{e^{2x} - 1}\, dx$
Ex 3.2(A) I (9) $\int \sin^4 x \cdot \cos^3 x\, dx$
Ex 3.2(A) I (10) $\int \frac{1}{4x + 5x^{-11}}\, dx$
Ex 3.2(A) I (11) $\int x^9 \cdot \sec^2 (x^{10})\, dx$
Ex 3.2(A) I (12) $\int e^{3 \log x} \cdot (x^4 + 1)^{-1}\, dx$
Ex 3.2(A) I (13) $\int \frac{\sqrt{\tan x}}{\sin x \cdot \cos x}\, dx$
Ex 3.2(A) I (14) $\int \frac{(x - 1)^2}{(x^2 + 1)^2}\, dx$
Ex 3.2(A) I (15) $\int \frac{2 \sin x \cdot \cos x}{3 \cos^2 x + 4 \sin^2 x}\, dx$
Ex 3.2(A) I (16) $\int \frac{1}{\sqrt{x} + \sqrt{x^3}}\, dx$
Ex 3.2(A) I (17) $\int \frac{10 x^9 + 10^x \cdot \log 10}{10^x + x^{10}}\, dx$
Ex 3.2(A) I (18) $\int \frac{x^{n-1}}{\sqrt{1 + 4x^n}}\, dx$
Ex 3.2(A) I (19) $\int (2x + 1)\sqrt{x + 2}\, dx$
Ex 3.2(A) I (20) $\int x^5 \cdot \sqrt{a^2 + x^2}\, dx$
Ex 3.2(A) I (21) $\int (5 - 3x)(2 - 3x)^{-\frac{1}{2}}\, dx$
Ex 3.2(A) I (22) $\int \frac{7 + 4x + 5x^2}{(2x + 3)^{\frac{3}{2}}}\, dx$
Ex 3.2(A) I (23) $\int \frac{x^2}{\sqrt{9 - x^6}}\, dx$
Ex 3.2(A) I (24) $\int \frac{1}{x (x^3 - 1)}\, dx$
Ex 3.2(A) I (25) $\int \frac{1}{x \cdot \log x \cdot \log (\log x)}\, dx$
Integrate the following functions w. r. t. x :
Ex 3.2(A) II (1) $\int \frac{\cos 3x - \cos 4x}{\sin 3x + \sin 4x}\, dx$
Ex 3.2(A) II (2) $\int \frac{\cos x}{\sin (x - a)}\, dx$
Ex 3.2(A) II (3) $\int \frac{\sin (x - a)}{\cos (x + b)}\, dx$
Ex 3.2(A) II (4) $\int \frac{1}{\sin x \cdot \cos x + 2 \cos^2 x}\, dx$
Ex 3.2(A) II (5) $\int \frac{\sin x + 2 \cos x}{3 \sin x + 4 \cos x}\, dx$
Ex 3.2(A) II (6) $\int \frac{1}{2 + 3 \tan x}\, dx$
Ex 3.2(A) II (7) $\int \frac{4 e^x - 25}{2 e^x - 5}\, dx$
Ex 3.2(A) II (8) $\int \frac{20 + 12 e^x}{3 e^x + 4}\, dx$
Ex 3.2(A) II (9) $\int \frac{3 e^{2x} + 5}{4 e^{2x} - 5}\, dx$
Ex 3.2(A) II (10) $\int \cos^8 x \cdot \cot x\, dx$
Ex 3.2(A) II (11) $\int \tan^5 x\, dx$
Ex 3.2(A) II (12) $\int \cos^7 x\, dx$
Ex 3.2(A) II (13) $\int \tan 3x \cdot \tan 2x \cdot \tan x\, dx$
Ex 3.2(A) II (14) $\int \sin^5 x \cdot \cos^8 x\, dx$
Ex 3.2(A) II (15) $\int 3^{\cos^2 x} \cdot \sin 2x\, dx$
Ex 3.2(A) II (16) $\int \frac{\sin 6x}{\sin 10x \cdot \sin 4x}\, dx$
Ex 3.2(A) II (17) $\int \frac{\sin x \cdot \cos^3 x}{1 + \cos^2 x}\, dx$

3.2B SolvedEx.1 $\int \frac{1}{4x^2 + 11}\, dx$
3.2B SolvedEx.2 $\int \frac{1}{a^2 - b^2 x^2}\, dx$
3.2B SolvedEx.3 $\int \frac{1}{\sqrt{3x^2 - 7}}\, dx$
3.2B SolvedEx.4 $\int \frac{1}{x^2 + 8x + 12}\, dx$
3.2B SolvedEx.5 $\int \frac{1}{\sqrt{3x^2 - 4x + 2}}\, dx$
3.2B SolvedEx.6 $\int \frac{1}{3 - 10x - 25x^2}\, dx$
3.2B SolvedEx.7 $\int \frac{1}{\sqrt{1 + x - x^2}}\, dx$
3.2B SolvedEx.8 $\int \frac{\sin 2x}{3\sin^4 x - 4\sin^2 x + 1}\, dx$
3.2B SolvedEx.9 $\int \frac{e^{\frac{x}{2}}}{\sqrt{e^{-x} - e^{x}}}\, dx$
3.2B SolvedEx.10 $\int \left(\sqrt{\tan x} + \sqrt{\cot x}\right) dx$
3.2B SolvedEx.11 $\int \frac{1}{5 - 4\cos x}\, dx$
3.2B SolvedEx.12 $\int \frac{1}{2 - 3\sin 2x}\, dx$
3.2B SolvedEx.13 $\int \frac{1}{3 - 2\sin x + 5\cos x}\, dx$
3.2B SolvedEx.14 $\int \frac{1}{\sin x - \sqrt{3}\cos x}\, dx$
3.2B SolvedEx.15 $\int \frac{1}{3 + 2\sin^2 x + 5\cos^2 x}\, dx$
3.2B SolvedEx.16 $\int \frac{\cos\theta}{\cos 3\theta}\, d\theta$

Evaluate the following :
Ex 3.2(B) I (i) $\int \frac{1}{4x^2 - 3} \, dx$
Ex 3.2(B) I (ii) $\int \frac{1}{25 - 9x^2} \, dx$
Ex 3.2(B) I (iii) $\int \frac{1}{7 + 2x^2} \, dx$
Ex 3.2(B) I (iv) $\int \frac{1}{\sqrt{3x^2 + 8}} \, dx$
Ex 3.2(B) I (v) $\int \frac{1}{\sqrt{11 - 4x^2}} \, dx$
Ex 3.2(B) I (vi) $\int \frac{1}{\sqrt{2x^2 - 5}} \, dx$
Ex 3.2(B) I (vii) $\int \sqrt{\frac{9 + x}{9 - x}} \, dx$
Ex 3.2(B) I (viii) $\int \sqrt{\frac{2 + x}{2 - x}} \, dx$
Ex 3.2(B) I (ix) $\int \sqrt{\frac{10 + x}{10 - x}} \, dx$
Ex 3.2(B) I (x) $\int \frac{1}{x^2 + 8x + 12} \, dx$
Ex 3.2(B) I (xi) $\int \frac{1}{1 + x - x^2} \, dx$
Ex 3.2(B) I (xii) $\int \frac{1}{4x^2 - 20x + 17} \, dx$
Ex 3.2(B) I (xiii) $\int \frac{1}{5 - 4x - 3x^2} \, dx$
Ex 3.2(B) I (xiv) $\int \frac{1}{\sqrt{3x^2 + 5x + 7}} \, dx$
Ex 3.2(B) I (xv) $\int \frac{1}{\sqrt{x^2 + 8x - 20}} \, dx$
Ex 3.2(B) I (xvi) $\int \frac{1}{\sqrt{8 - 3x + 2x^2}} \, dx$
Ex 3.2(B) I (xvii) $\int \frac{1}{\sqrt{(x - 3)(x + 2)}} \, dx$
Ex 3.2(B) I (xviii) $\int \frac{1}{4 + 3\cos^2 x} \, dx$
Ex 3.2(B) I (xix) $\int \frac{1}{\cos 2x + 3\sin^2 x} \, dx$
Ex 3.2(B) I (xx) $\int \frac{\sin x}{\sin 3x} \, dx$
Integrate the following functions w. r. t. x :
Ex 3.2(B) II (i) $\int \frac{1}{3 + 2\sin x} \, dx$
Ex 3.2(B) II (ii) $\int \frac{1}{4 - 5\cos x} \, dx$
Ex 3.2(B) II (iii) $\int \frac{1}{2 + \cos x - \sin x} \, dx$
Ex 3.2(B) II (iv) $\int \frac{1}{3 + 2\sin x - \cos x} \, dx$
Ex 3.2(B) II (v) $\int \frac{1}{3 - 2\cos 2x} \, dx$
Ex 3.2(B) II (vi) $\int \frac{1}{2\sin 2x - 3} \, dx$
Ex 3.2(B) II (vii) $\int \frac{1}{3 + 2\sin 2x + 4\cos 2x} \, dx$
Ex 3.2(B) II (viii) $\int \frac{1}{\cos x - \sin x} \, dx$
Ex 3.2(B) II (ix) $\int \frac{1}{\cos x - \sqrt{3}\,\sin x} \, dx$

3.2C SolvedEx.1 $\int \frac{2x-3}{3x^2+4x+5}\, dx$
3.2C SolvedEx.2 $\int \sqrt{\frac{x-5}{x-7}}\, dx$
3.2C SolvedEx.3 $\int \sqrt{\frac{8-x}{x}}\, dx$

Evaluate :
Ex 3.2(C) I (i) $\int \frac{3x+4}{x^2+6x+5}\, dx$
Ex 3.2(C) I (ii) $\int \frac{2x+1}{x^2+4x-5}\, dx$
Ex 3.2(C) I (iii) $\int \frac{2x+3}{2x^2+3x-1}\, dx$
Ex 3.2(C) I (iv) $\int \frac{3x+4}{\sqrt{2x^2+2x+1}}\, dx$
Ex 3.2(C) I (v) $\int \frac{7x+3}{\sqrt{3+2x-x^2}}\, dx$
Ex 3.2(C) I (vi) $\int \sqrt{\frac{x-7}{x-9}}\, dx$
Ex 3.2(C) I (vii) $\int \sqrt{\frac{9-x}{x}}\, dx$
Ex 3.2(C) I (viii) $\int \frac{3\cos x}{4\sin^2 x + 4\sin x - 1}\, dx$
Ex 3.2(C) I (ix) $\int \sqrt{\frac{e^{3x} - e^{2x}}{e^x + 1}}\, dx$

3.3 SolvedEx.1 $\int x^2\, 5^x\, dx$
3.3 SolvedEx.2 $\int x \tan^{-1} x\, dx$
3.3 SolvedEx.3 $\int \frac{x}{1 - \sin x}\, dx$
3.3 SolvedEx.4 $\int e^{2x} \sin 3x\, dx$
3.3 SolvedEx.5 $\int \left[ \log (\log x) + \frac{1}{(\log x)^2} \right] dx$
3.3 SolvedEx.6 $\int \sqrt{a^2 - x^2}\, dx$
3.3 SolvedEx.7 $\int \sqrt{a^2 + x^2}\, dx$
3.3 SolvedEx.8 $\int \sqrt{x^2 - a^2}\, dx$
3.3 SolvedEx.9 $\int x \sin^{-1} x\, dx$
3.3 SolvedEx.10 $\int \cos^{-1} \sqrt{x}\, dx$
3.3 SolvedEx.11 $\int \sqrt{4 + 3x - 2x^2}\, dx$
3.3.3 SolvedEx.1 $\int e^x \left( \frac{2 + \sin 2x}{1 + \cos 2x} \right) dx$
3.3.3 SolvedEx.2 $\int e^x \left[ \frac{x + 2}{(x + 3)^2} \right] dx$
3.3.3 SolvedEx.3 $\int e^{\tan^{-1} x} \left( \frac{1 + x + x^2}{1 + x^2} \right) dx$
3.3.3 SolvedEx.4 $\int \frac{(x^2 + 1) \cdot e^x}{(x + 1)^2}\, dx$

Evaluate the following :
Ex 3.3 I (i) $\int x^2 \cdot \log x\, dx$
Ex 3.3 I (ii) $\int x^2 \cdot \sin 3x\, dx$
Ex 3.3 I (iii) $\int x \cdot \tan^{-1} x\, dx$
Ex 3.3 I (iv) $\int x^2 \cdot \tan^{-1} x\, dx$
Ex 3.3 I (v) $\int x^3 \cdot \tan^{-1} x\, dx$
Ex 3.3 I (vi) $\int (\log x)^2\, dx$
Ex 3.3 I (vii) $\int \sec^3 x\, dx$
Ex 3.3 I (viii) $\int x \cdot \sin^2 x\, dx$
Ex 3.3 I (ix) $\int x^3 \cdot \log x\, dx$
Ex 3.3 I (x) $\int e^{2x} \cdot \cos 3x\, dx$
Ex 3.3 I (xi) $\int x \cdot \sin^{-1} x\, dx$
Ex 3.3 I (xii) $\int x^2 \cdot \cos^{-1} x\, dx$
Ex 3.3 I (xiii) $\int \frac{\log (\log x)}{x}\, dx$
Ex 3.3 I (xiv) $\int \frac{t \sin^{-1} t}{\sqrt{1 - t^2}}\, dt$
Ex 3.3 I (xv) $\int \cos \sqrt{x}\, dx$
Ex 3.3 I (xvi) $\int \sin \theta \cdot \log (\cos \theta)\, d\theta$
Ex 3.3 I (xvii) $\int x \cdot \cos^3 x\, dx$
Ex 3.3 I (xviii) $\int \frac{\sin (\log x)^2}{x} \cdot \log x\, dx$
Ex 3.3 I (xix) $\int \frac{\log x}{x}\, dx$
Ex 3.3 I (xx) $\int x \cdot \sin 2x \cdot \cos 5x\, dx$
Ex 3.3 I (xxi) $\int \cos \left(\sqrt[3]{x}\right)\, dx$
Integrate the following functions w. r. t. x :
Ex 3.3 II (i) $\int e^{2x} \cdot \sin 3x\, dx$
Ex 3.3 II (ii) $\int e^{-x} \cdot \cos 2x\, dx$
Ex 3.3 II (iii) $\int \sin (\log x)\, dx$
Ex 3.3 II (iv) $\int \sqrt{5x^2 + 3}\, dx$
Ex 3.3 II (v) $\int x^2 \cdot \sqrt{a^2 - x^6}\, dx$
Ex 3.3 II (vi) $\int \sqrt{(x - 3)(7 - x)}\, dx$
Ex 3.3 II (vii) $\int \sqrt{4^x (4^x + 4)}\, dx$
Ex 3.3 II (viii) $\int (x + 1)\sqrt{2x^2 + 3}\, dx$
Ex 3.3 II (ix) $\int x \sqrt{5 - 4x - x^2}\, dx$
Ex 3.3 II (x) $\int \sec^2 x \cdot \sqrt{\tan^2 x + \tan x - 7}\, dx$
Ex 3.3 II (xi) $\int \sqrt{x^2 + 2x + 5}\, dx$
Ex 3.3 II (xii) $\int \sqrt{2x^2 + 3x + 4}\, dx$
Integrate the following functions w. r. t. x :
Ex 3.3 III (i) $\int (2 + \cot x - \operatorname{cosec}^2 x) \cdot e^x\, dx$
Ex 3.3 III (ii) $\int \left(\frac{1 + \sin x}{1 + \cos x}\right) \cdot e^x\, dx$
Ex 3.3 III (iii) $\int e^x \cdot \left(\frac{1}{x} - \frac{1}{x^2}\right)\, dx$
Ex 3.3 III (iv) $\int \left(\frac{x}{(x + 1)^2}\right) \cdot e^x\, dx$
Ex 3.3 III (v) $\int \frac{e^x}{x} \left[x (\log x)^2 + 2 (\log x)\right]\, dx$
Ex 3.3 III (vi) $\int e^{5x} \cdot \left(\frac{5x \cdot \log x + 1}{x}\right)\, dx$
Ex 3.3 III (vii) $\int e^{\sin^{-1} x} \cdot \left(\frac{x + \sqrt{1 - x^2}}{\sqrt{1 - x^2}}\right)\, dx$
Ex 3.3 III (viii) $\int \log (1 + x)^{(1 + x)}\, dx$
Ex 3.3 III (ix) $\int \operatorname{cosec} (\log x) \left[1 - \cot (\log x)\right]\, dx$

3.4 SolvedEx.1 $\int \frac{3x^2 + 4x - 5}{(x^2 - 1)(x + 2)}\, dx$
3.4 SolvedEx.2 $\int \frac{2x^2 - 3}{(x^2 - 5)(x^2 + 4)}\, dx$
3.4 SolvedEx.3 $\int \frac{1}{(\sin \theta)(3 + 2 \cos \theta)}\, d\theta$
3.4 SolvedEx.4 $\int \frac{1}{2 \cos x + \sin 2x}\, dx$
3.4 SolvedEx.5 $\int \frac{\tan \theta + \tan^3 \theta}{1 + \tan^3 \theta}\, d\theta$

Integrate the following w. r. t. x :
Ex 3.4 I (i) $\int \frac{x^2 + 2}{(x - 1)(x + 2)(x + 3)} \, dx$
Ex 3.4 I (ii) $\int \frac{x^2}{(x^2 + 1)(x^2 - 2)(x^2 + 3)} \, dx$
Ex 3.4 I (iii) $\int \frac{12x + 3}{6x^2 + 13x - 63} \, dx$
Ex 3.4 I (iv) $\int \frac{2x}{4 - 3x - x^2} \, dx$
Ex 3.4 I (v) $\int \frac{x^2 + x - 1}{x^2 + x - 6} \, dx$
Ex 3.4 I (vi) $\int \frac{6x^3 + 5x^2 - 7}{3x^2 - 2x - 1} \, dx$
Ex 3.4 I (vii) $\int \frac{12x^2 - 2x - 9}{(4x^2 - 1)(x + 3)} \, dx$
Ex 3.4 I (viii) $\int \frac{1}{x(x^5 + 1)} \, dx$
Ex 3.4 I (ix) $\int \frac{2x^2 - 1}{x^4 + 9x^2 + 20} \, dx$
Ex 3.4 I (x) $\int \frac{x^2 + 3}{(x^2 - 1)(x^2 - 2)} \, dx$
Ex 3.4 I (xi) $\int \frac{2x}{(2 + x^2)(3 + x^2)} \, dx$
Ex 3.4 I (xii) $\int \frac{2^x}{4^x - 3 \cdot 2^x - 4} \, dx$
Ex 3.4 I (xiii) $\int \frac{3x - 2}{(x + 1)^2 (x + 3)} \, dx$
Ex 3.4 I (xiv) $\int \frac{5x^2 + 20x + 6}{x^3 + 2x^2 + x} \, dx$
Ex 3.4 I (xv) $\int \frac{1}{x(1 + 4x^3 + 3x^6)} \, dx$
Ex 3.4 I (xvi) $\int \frac{1}{x^3 - 1} \, dx$
Ex 3.4 I (xvii) $\int \frac{(3 \sin x - 2) \cdot \cos x}{5 - 4 \sin x - \cos^2 x} \, dx$
Ex 3.4 I (xviii) $\int \frac{1}{\sin x + \sin 2x} \, dx$
Ex 3.4 I (xix) $\int \frac{1}{2 \sin x + \sin 2x} \, dx$
Ex 3.4 I (xx) $\int \frac{1}{\sin 2x + \cos x} \, dx$
Ex 3.4 I (xxi) $\int \frac{1}{\sin x \cdot (3 + 2 \cos x)} \, dx$
Ex 3.4 I (xxii) $\int \frac{5 \cdot e^x}{(e^x + 1)(e^{2x} + 9)} \, dx$
Ex 3.4 I (xxiii) $\int \frac{2 \log x + 3}{x(3 \log x + 2)[(\log x)^2 + 1]} \, dx$

Misc I Q.1 $\int \frac{1+x+\sqrt{x+x^2}}{\sqrt{x}+\sqrt{1+x}}\, dx =$
1. A. $\frac{1}{2}\sqrt{x+1}+c$
2. B. $\frac{2}{3}(x+1)^{3/2}+c$
3. C. $\sqrt{x+1}+c$
4. D. $2(x+1)^{3/2}+c$
Misc I Q.2 $\int \frac{1}{x+x^5}\, dx = f(x)+c,$
1. A. $\log x - f(x)+c$
2. B. $f(x)+\log x+c$
3. C. $f(x)-\log x+c$
4. D. $\frac{1}{5}x^5 f(x)+c$
Misc I Q.3 $\int \frac{\log(3x)}{x\log(9x)}\, dx =$
1. A. $\log(3x)-\log(9x)+c$
2. B. $\log(x)-(\log 3)\cdot\log(\log 9x)+c$
3. C. $\log 9-(\log x)\cdot\log(\log 3x)+c$
4. D. $\log(x)+(\log 3)\cdot\log(\log 9x)+c$
Misc I Q.4 $\int \frac{\sin^m x}{\cos^{m+2}x}\, dx =$
1. A. $\frac{\tan^{m+1}x}{m+1}+c$
2. B. $(m+2)\tan^{m+1}x+c$
3. C. $\frac{\tan^m x}{m}+c$
4. D. $(m+1)\tan^{m+1}x+c$
Misc I Q.5 $\int \tan(\sin^{-1}x)\, dx =$
1. A. $(1-x^2)^{-\frac{1}{2}}+c$
2. B. $(1-x^2)^{\frac{1}{2}}+c$
3. C. $\frac{\tan^m x}{\sqrt{1-x^2}}+c$
4. D. $-\sqrt{1-x^2}+c$
Misc I Q.6 $\int \frac{x-\sin x}{1-\cos x}\, dx =$
1. A. $x\cot\left(\frac{x}{2}\right)+c$
2. B. $-x\cot\left(\frac{x}{2}\right)+c$
3. C. $\cot\left(\frac{x}{2}\right)+c$
4. D. $x\tan\left(\frac{x}{2}\right)+c$
Misc I Q.7 $f(x)=\frac{\sin^{-1}x}{\sqrt{1-x^2}},\ g(x)=e^{\sin^{-1}x},$
1. A. $e^{\sin^{-1}x}(\sin^{-1}x-1)+c$
2. B. $e^{\sin^{-1}x}(1-\sin^{-1}x)+c$
3. C. $e^{\sin^{-1}x}(\sin^{-1}x+1)+c$
4. D. $e^{\sin^{-1}x}(\sin^{-1}x-1)+c$
Misc I Q.8 $\int \tan^3 x\sec^3 x\, dx = \left(\frac{1}{m}\right)\sec^m x - \left(\frac{1}{n}\right)\sec^n x + c,$
1. A. $(5, 3)$
2. B. $(3, 5)$
3. C. $\left(\frac{1}{5},\frac{1}{3}\right)$
4. D. $(4, 4)$
Misc I Q.9 $\int \frac{1}{\cos x-\cos^2 x}\, dx =$
1. A. $\log(\csc x-\cot x)+\tan\left(\frac{x}{2}\right)+c$
2. B. $\sin 2x-\cos x+c$
3. C. $\log(\sec x+\tan x)-\cot\left(\frac{x}{2}\right)+c$
4. D. $\cos 2x-\sin x+c$
Misc I Q.10 $\int \frac{\sqrt{\cot x}}{\sin x\,\cos x}\, dx =$
1. A. $2\sqrt{\cot x}+c$
2. B. $-2\sqrt{\cot x}+c$
3. C. $\frac{1}{2}\sqrt{\cot x}+c$
4. D. $\sqrt{\cot x}+c$
Misc I Q.11 $\int \frac{e^x(x-1)}{x^2}\, dx =$
1. A. $\frac{e^x}{x}+c$
2. B. $\frac{e^x}{x^2}+c$
3. C. $\left(x-\frac{1}{x}\right)e^x+c$
4. D. $xe^{-x}+c$
Misc I Q.12 $\int \sin(\log x)\, dx =$
1. A. $\frac{x}{2}[\sin(\log x)-\cos(\log x)]+c$
2. B. $\frac{x}{2}[\sin(\log x)+\cos(\log x)]+c$
3. C. $\frac{x}{2}[\cos(\log x)-\sin(\log x)]+c$
4. D. $\frac{x}{4}[\cos(\log x)-\sin(\log x)]+c$
Misc I Q.13 $\int x^x(1+\log x)\, dx =$
1. A. $\frac{1}{2}(1+\log x)^2+c$
2. B. $x^{2x}+c$
3. C. $x^x\log x+c$
4. D. $x^x+c$
Misc I Q.14 $\int \cos^{-\frac{3}{7}}x\,\sin^{-\frac{11}{7}}x\, dx =$
1. A. $\log\left(\sin^{-\frac{4}{7}}x\right)+c$
2. B. $\frac{4}{7}\tan^{\frac{4}{7}}x+c$
3. C. $-\frac{7}{4}\tan^{-\frac{4}{7}}x+c$
4. D. $\log\left(\cos^{\frac{3}{7}}x\right)+c$
Misc I Q.15 $2\int \frac{\cos^2 x-\sin^2 x}{\cos^2 x+\sin^2 x}\, dx =$
1. A. $\sin 2x+c$
2. B. $\cos 2x+c$
3. C. $\tan 2x+c$
4. D. $2\sin 2x+c$
Misc I Q.16 $\int \frac{dx}{\cos x\sqrt{\sin^2 x-\cos^2 x}} =$
1. A. $\log\left(\tan x-\sqrt{\tan^2 x-1}\right)+c$
2. B. $\sin^{-1}(\tan x)+c$
3. C. $1+\sin^{-1}(\cot x)+c$
4. D. $\log\left(\tan x+\sqrt{\tan^2 x-1}\right)+c$
Misc I Q.17 $\int \frac{\log x}{(\log ex)^2}\, dx =$
1. A. $\frac{x}{1+\log x}+c$
2. B. $x(1+\log x)+c$
3. C. $\frac{1}{1+\log x}+c$
4. D. $\frac{1}{1-\log x}+c$
Misc I Q.18 $\int [\sin(\log x)+\cos(\log x)]\, dx =$
1. A. $x\cos(\log x)+c$
2. B. $\sin(\log x)+c$
3. C. $\cos(\log x)+c$
4. D. $x\sin(\log x)+c$
Misc I Q.19 $\int \frac{\cos 2x-1}{\cos 2x+1}\, dx =$
1. A. $\tan x-x+c$
2. B. $x+\tan x+c$
3. C. $x-\tan x+c$
4. D. $-x-\cot x+c$
Misc I Q.20 $\int \frac{e^{2x}+e^{-2x}}{e^x}\, dx =$
1. A. $e^x-\frac{1}{3e^{3x}}+c$
2. B. $e^x+\frac{1}{3e^{3x}}+c$
3. C. $e^{-x}+\frac{1}{3e^{3x}}+c$
4. D. $e^{-x}-\frac{1}{3e^{3x}}+c$

Misc II Q.1 $x$
Misc II Q.2 $x$
Misc II Q.3 $x$
Misc II Q.4 $t$
Misc II Q.5 $x$
Misc II Q.6 $\theta$
Misc II Q.7 $x$
Misc II Q.8 $x$
Misc II Q.9 $x$

Misc III Q.1 $\ \dfrac{(1+\log x)^3}{x}$
Misc III Q.2 $\ \cot^{-1}(1-x+x^2)$
Misc III Q.3 $\ \dfrac{1}{x\cdot\sin^2(\log x)}$
Misc III Q.4 $\ \sqrt{x}\,\sec\left(x^{\frac{3}{2}}\right)\cdot\tan\left(x^{\frac{3}{2}}\right)$
Misc III Q.5 $\ \log(1+\cos x) - x\cdot\tan\left(\dfrac{x}{2}\right)$
Misc III Q.6 $\ \dfrac{x^2}{\sqrt{1-x^6}}$
Misc III Q.7 $\ \dfrac{1}{(1-\cos 4x)(3-\cot 2x)}$
Misc III Q.8 $\ \log(\log x) + (\log x)^{-2}$
Misc III Q.9 $\ \dfrac{1}{2\cos x+3\sin x}$
Misc III Q.10 $\ \dfrac{1}{x^3\cdot\sqrt{x^2-1}}$
Misc III Q.11 $\ \dfrac{3x+1}{\sqrt{-2x^2+x+3}}$
Misc III Q.12 $\ \log(x^2+1)$
Misc III Q.13 $\ e^{2x}\cdot\sin x\cdot\cos x$
Misc III Q.14 $\ \dfrac{x^2}{(x-1)(3x-1)(3x-2)}$
Misc III Q.15 $\ \dfrac{1}{\sin x+\sin 2x}$
Misc III Q.16 $\ \sec^2 x\cdot\sqrt{7+2\tan x-\tan^2 x}$
Misc III Q.17 $\ \dfrac{x+5}{x^3+3x^2-x-3}$
Misc III Q.18 $\ \dfrac{1}{x\cdot(x^5+1)}$
Misc III Q.19 $\ \dfrac{\sqrt{\tan x}}{\sin x\cdot\cos x}$
Misc III Q.20 $\ \sec^4 x\cdot\csc^2 x$

3.1 Elementary Integration

Solved Examples

Exercise 3.1

3.2 Methods of Integration — Substitution and Trigonometric

Solved Examples

Exercise 3.2 (A)

3.2 Special Integrals

Solved Examples

Exercise 3.2 (B)

3.2 Integrals of the Type (px+q) over a Quadratic

Solved Examples

Exercise 3.2 (C)

3.3 Integration by Parts

Solved Examples

Exercise 3.3

3.4 Integration by Partial Fraction

Solved Examples

Exercise 3.4

Miscellaneous Exercise 3

Choose the correct option

Integrate / Evaluate

Integrate the following