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Exam: CBSE Class 12
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Q276
#276
Mathematics → Integrals → Integration by Substitution
·
Moderate
Add
Integrate the function
1
x
+
a
+
x
+
b
\frac{1}{\sqrt{x+a} + \sqrt{x+b}}
x
+
a
+
x
+
b
1
.
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[QMisc Q2 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q277
#277
Mathematics → Integrals → Integration by Substitution
·
Hard
Add
Integrate the function
1
x
a
x
−
x
2
\frac{1}{x\sqrt{ax - x^2}}
x
a
x
−
x
2
1
. [Hint: Put
x
=
a
t
x = \frac{a}{t}
x
=
t
a
]
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[QMisc Q3 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q278
#278
Mathematics → Integrals → Integration by Substitution
·
Hard
Add
Integrate the function
1
x
2
(
x
4
+
1
)
3
4
\frac{1}{x^2 (x^4 + 1)^{\frac{3}{4}}}
x
2
(
x
4
+
1
)
4
3
1
.
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[QMisc Q4 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q279
#279
Mathematics → Integrals → Integration by Substitution
·
Hard
Add
Integrate the function
1
x
1
2
+
x
1
3
\frac{1}{x^{\frac{1}{2}} + x^{\frac{1}{3}}}
x
2
1
+
x
3
1
1
. [Hint:
1
x
1
2
+
x
1
3
=
1
x
1
3
(
1
+
x
1
6
)
\frac{1}{x^{\frac{1}{2}} + x^{\frac{1}{3}}} = \frac{1}{x^{\frac{1}{3}}\left(1 + x^{\frac{1}{6}}\right)}
x
2
1
+
x
3
1
1
=
x
3
1
(
1
+
x
6
1
)
1
, put
x
=
t
6
x = t^6
x
=
t
6
]
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[QMisc Q5 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q280
#280
Mathematics → Integrals → Integration by Partial Fractions
·
Moderate
Add
Integrate the function
5
x
(
x
+
1
)
(
x
2
+
9
)
\frac{5x}{(x+1)(x^2+9)}
(
x
+
1
)
(
x
2
+
9
)
5
x
.
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[QMisc Q6 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q281
#281
Mathematics → Integrals → Integration using Trigonometric Identities
·
Moderate
Add
Integrate the function
sin
x
sin
(
x
−
a
)
\frac{\sin x}{\sin(x - a)}
s
i
n
(
x
−
a
)
s
i
n
x
.
Show model answer
[QMisc Q7 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q282
#282
Mathematics → Integrals → Integration as the Inverse of Differentiation
·
Moderate
Add
Integrate the function
e
5
log
x
−
e
4
log
x
e
3
log
x
−
e
2
log
x
\frac{e^{5\log x} - e^{4\log x}}{e^{3\log x} - e^{2\log x}}
e
3
l
o
g
x
−
e
2
l
o
g
x
e
5
l
o
g
x
−
e
4
l
o
g
x
.
Show model answer
[QMisc Q8 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q283
#283
Mathematics → Integrals → Integrals of Some Particular Functions
·
Easy
Add
Integrate the function
cos
x
4
−
sin
2
x
\frac{\cos x}{\sqrt{4 - \sin^2 x}}
4
−
s
i
n
2
x
c
o
s
x
.
Show model answer
[QMisc Q9 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q284
#284
Mathematics → Integrals → Integration using Trigonometric Identities
·
Hard
Add
Integrate the function
sin
8
x
−
cos
8
x
1
−
2
sin
2
x
cos
2
x
\frac{\sin^8 x - \cos^8 x}{1 - 2\sin^2 x \cos^2 x}
1
−
2
s
i
n
2
x
c
o
s
2
x
s
i
n
8
x
−
c
o
s
8
x
.
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[QMisc Q10 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q285
#285
Mathematics → Integrals → Integration using Trigonometric Identities
·
Moderate
Add
Integrate the function
1
cos
(
x
+
a
)
cos
(
x
+
b
)
\frac{1}{\cos(x + a)\cos(x + b)}
c
o
s
(
x
+
a
)
c
o
s
(
x
+
b
)
1
.
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[QMisc Q11 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
Report
Q286
#286
Mathematics → Integrals → Integrals of Some Particular Functions
·
Moderate
Add
Integrate the function
x
3
1
−
x
8
\frac{x^3}{\sqrt{1 - x^8}}
1
−
x
8
x
3
.
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[QMisc Q12 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q287
#287
Mathematics → Integrals → Integration by Partial Fractions
·
Moderate
Add
Integrate the function
e
x
(
1
+
e
x
)
(
2
+
e
x
)
\frac{e^x}{(1 + e^x)(2 + e^x)}
(
1
+
e
x
)
(
2
+
e
x
)
e
x
.
Show model answer
[QMisc Q13 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q288
#288
Mathematics → Integrals → Integration by Partial Fractions
·
Moderate
Add
Integrate the function
1
(
x
2
+
1
)
(
x
2
+
4
)
\frac{1}{(x^2 + 1)(x^2 + 4)}
(
x
2
+
1
)
(
x
2
+
4
)
1
.
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[QMisc Q14 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q289
#289
Mathematics → Integrals → Integration by Substitution
·
Moderate
Add
Integrate the function
cos
3
x
e
log
sin
x
\cos^3 x\; e^{\log \sin x}
cos
3
x
e
l
o
g
s
i
n
x
.
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[QMisc Q15 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
Report
Q290
#290
Mathematics → Integrals → Integration by Substitution
·
Moderate
Add
Integrate the function
e
3
log
x
(
x
4
+
1
)
−
1
e^{3\log x}(x^4 + 1)^{-1}
e
3
l
o
g
x
(
x
4
+
1
)
−
1
.
Show model answer
[QMisc Q16 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q291
#291
Mathematics → Integrals → Integration by Substitution
·
Moderate
Add
Integrate the function
f
′
(
a
x
+
b
)
[
f
(
a
x
+
b
)
]
n
f'(ax + b)\,[f(ax + b)]^n
f
′
(
a
x
+
b
)
[
f
(
a
x
+
b
)
]
n
.
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[QMisc Q17 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q292
#292
Mathematics → Integrals → Integration by Substitution
·
Hard
Add
Integrate the function
1
sin
3
x
sin
(
x
+
α
)
\frac{1}{\sqrt{\sin^3 x \, \sin(x + \alpha)}}
s
i
n
3
x
s
i
n
(
x
+
α
)
1
.
Show model answer
[QMisc Q18 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q293
#293
Mathematics → Integrals → Integration by Substitution
·
Hard
Add
Integrate the function
1
−
x
1
+
x
\sqrt{\frac{1 - \sqrt{x}}{1 + \sqrt{x}}}
1
+
x
1
−
x
.
Show model answer
[QMisc Q19 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q294
#294
Mathematics → Integrals → Integration by Parts
·
Moderate
Add
Integrate the function
2
+
sin
2
x
1
+
cos
2
x
e
x
\frac{2 + \sin 2x}{1 + \cos 2x}\, e^x
1
+
c
o
s
2
x
2
+
s
i
n
2
x
e
x
.
Show model answer
[QMisc Q20 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q295
#295
Mathematics → Integrals → Integration by Partial Fractions
·
Moderate
Add
Integrate the function
x
2
+
x
+
1
(
x
+
1
)
2
(
x
+
2
)
\frac{x^2 + x + 1}{(x + 1)^2 (x + 2)}
(
x
+
1
)
2
(
x
+
2
)
x
2
+
x
+
1
.
Show model answer
[QMisc Q21 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q296
#296
Mathematics → Integrals → Integration by Parts
·
Hard
Add
Integrate the function
tan
−
1
1
−
x
1
+
x
\tan^{-1}\sqrt{\frac{1 - x}{1 + x}}
tan
−
1
1
+
x
1
−
x
.
Show model answer
[QMisc Q22 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q297
#297
Mathematics → Integrals → Integration by Substitution
·
Hard
Add
Integrate the function
x
2
+
1
[
log
(
x
2
+
1
)
−
2
log
x
]
x
4
\frac{\sqrt{x^2 + 1}\,\left[\log(x^2 + 1) - 2\log x\right]}{x^4}
x
4
x
2
+
1
[
l
o
g
(
x
2
+
1
)
−
2
l
o
g
x
]
.
Show model answer
[QMisc Q23 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
Report
Q298
#298
Mathematics → Integrals → Integration by Parts
·
Hard
Add
Evaluate the definite integral
∫
π
2
π
e
x
(
1
−
sin
x
1
−
cos
x
)
d
x
\int_{\frac{\pi}{2}}^{\pi} e^x\left(\frac{1 - \sin x}{1 - \cos x}\right) dx
∫
2
π
π
e
x
(
1
−
c
o
s
x
1
−
s
i
n
x
)
d
x
.
Show model answer
[QMisc Q24 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
Report
Q299
#299
Mathematics → Integrals → Definite Integrals by Substitution
·
Hard
Add
Evaluate the definite integral
∫
0
π
4
sin
x
cos
x
cos
4
x
+
sin
4
x
d
x
\int_{0}^{\frac{\pi}{4}} \frac{\sin x \cos x}{\cos^4 x + \sin^4 x}\, dx
∫
0
4
π
c
o
s
4
x
+
s
i
n
4
x
s
i
n
x
c
o
s
x
d
x
.
Show model answer
[QMisc Q25 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q300
#300
Mathematics → Integrals → Definite Integrals by Substitution
·
Moderate
Add
Evaluate the definite integral
∫
0
π
2
cos
2
x
d
x
cos
2
x
+
4
sin
2
x
\int_{0}^{\frac{\pi}{2}} \frac{\cos^2 x\, dx}{\cos^2 x + 4\sin^2 x}
∫
0
2
π
c
o
s
2
x
+
4
s
i
n
2
x
c
o
s
2
x
d
x
.
Show model answer
[QMisc Q26 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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