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Exam: CBSE Class 12
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Q251
#251
Mathematics → Integrals → Definite Integrals and the Fundamental Theorem
·
Moderate
Add
Evaluate the definite integral
∫
0
2
6
x
+
3
x
2
+
4
d
x
\int_{0}^{2} \frac{6x + 3}{x^{2} + 4}\,dx
∫
0
2
x
2
+
4
6
x
+
3
d
x
.
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[QEx 7.8 Q19 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q252
#252
Mathematics → Integrals → Definite Integrals and the Fundamental Theorem
·
Moderate
Add
Evaluate the definite integral
∫
0
1
(
x
e
x
+
sin
π
x
4
)
d
x
\int_{0}^{1} \left(x\, e^{x} + \sin\frac{\pi x}{4}\right)\,dx
∫
0
1
(
x
e
x
+
sin
4
π
x
)
d
x
.
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[QEx 7.8 Q20 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q253
#253
Mathematics → Integrals → Definite Integrals and the Fundamental Theorem
·
Moderate
Add
∫
1
3
d
x
1
+
x
2
\int_{1}^{\sqrt{3}} \frac{dx}{1 + x^{2}}
∫
1
3
1
+
x
2
d
x
equals
A
π
3
\frac{\pi}{3}
3
π
B
2
π
3
\frac{2\pi}{3}
3
2
π
C
π
6
\frac{\pi}{6}
6
π
D
π
12
\frac{\pi}{12}
12
π
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[QEx 7.8 Q21 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q254
#254
Mathematics → Integrals → Definite Integrals and the Fundamental Theorem
·
Moderate
Add
∫
0
2
3
d
x
4
+
9
x
2
\int_{0}^{\frac{2}{3}} \frac{dx}{4 + 9x^{2}}
∫
0
3
2
4
+
9
x
2
d
x
equals
A
π
6
\frac{\pi}{6}
6
π
B
π
12
\frac{\pi}{12}
12
π
C
π
24
\frac{\pi}{24}
24
π
D
π
4
\frac{\pi}{4}
4
π
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[QEx 7.8 Q22 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q255
#255
Mathematics → Integrals → Definite Integrals by Substitution
·
Moderate
Add
Evaluate
∫
−
1
1
5
x
4
x
5
+
1
d
x
\int_{-1}^{1} 5x^{4}\sqrt{x^{5}+1}\,dx
∫
−
1
1
5
x
4
x
5
+
1
d
x
.
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[Q7.9 Eg.26 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q256
#256
Mathematics → Integrals → Definite Integrals by Substitution
·
Moderate
Add
Evaluate
∫
0
1
tan
−
1
x
1
+
x
2
d
x
\int_{0}^{1} \frac{\tan^{-1} x}{1+x^{2}}\,dx
∫
0
1
1
+
x
2
t
a
n
−
1
x
d
x
.
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[Q7.9 Eg.27 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q257
#257
Mathematics → Integrals → Definite Integrals by Substitution
·
Easy
Add
Evaluate the integral using substitution:
∫
0
1
x
x
2
+
1
d
x
\int_{0}^{1} \frac{x}{x^{2}+1}\,dx
∫
0
1
x
2
+
1
x
d
x
.
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[QEx 7.9 Q1 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q258
#258
Mathematics → Integrals → Definite Integrals by Substitution
·
Moderate
Add
Evaluate the integral using substitution:
∫
0
π
/
2
sin
ϕ
cos
5
ϕ
d
ϕ
\int_{0}^{\pi/2} \sqrt{\sin\phi}\,\cos^{5}\phi\,d\phi
∫
0
π
/2
sin
ϕ
cos
5
ϕ
d
ϕ
.
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[QEx 7.9 Q2 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q259
#259
Mathematics → Integrals → Definite Integrals by Substitution
·
Hard
Add
Evaluate the integral using substitution:
∫
0
1
sin
−
1
(
2
x
1
+
x
2
)
d
x
\int_{0}^{1} \sin^{-1}\left(\frac{2x}{1+x^{2}}\right)dx
∫
0
1
sin
−
1
(
1
+
x
2
2
x
)
d
x
.
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[QEx 7.9 Q3 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q260
#260
Mathematics → Integrals → Definite Integrals by Substitution
·
Moderate
Add
Evaluate the integral using substitution:
∫
0
2
x
x
+
2
d
x
\int_{0}^{2} x\sqrt{x+2}\,dx
∫
0
2
x
x
+
2
d
x
(Put
x
+
2
=
t
2
x+2 = t^{2}
x
+
2
=
t
2
).
Show model answer
[QEx 7.9 Q4 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q261
#261
Mathematics → Integrals → Definite Integrals by Substitution
·
Moderate
Add
Evaluate the integral using substitution:
∫
0
π
/
2
sin
x
1
+
cos
2
x
d
x
\int_{0}^{\pi/2} \frac{\sin x}{1+\cos^{2} x}\,dx
∫
0
π
/2
1
+
c
o
s
2
x
s
i
n
x
d
x
.
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[QEx 7.9 Q5 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q262
#262
Mathematics → Integrals → Definite Integrals by Substitution
·
Moderate
Add
Evaluate the integral using substitution:
∫
0
2
d
x
x
+
4
−
x
2
\int_{0}^{2} \frac{dx}{x+4-x^{2}}
∫
0
2
x
+
4
−
x
2
d
x
.
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[QEx 7.9 Q6 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q263
#263
Mathematics → Integrals → Definite Integrals by Substitution
·
Moderate
Add
Evaluate the integral using substitution:
∫
−
1
1
d
x
x
2
+
2
x
+
5
\int_{-1}^{1} \frac{dx}{x^{2}+2x+5}
∫
−
1
1
x
2
+
2
x
+
5
d
x
.
Show model answer
[QEx 7.9 Q7 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q264
#264
Mathematics → Integrals → Definite Integrals by Substitution
·
Hard
Add
Evaluate the integral using substitution:
∫
1
2
(
1
x
−
1
2
x
2
)
e
2
x
d
x
\int_{1}^{2} \left(\frac{1}{x} - \frac{1}{2x^{2}}\right) e^{2x}\,dx
∫
1
2
(
x
1
−
2
x
2
1
)
e
2
x
d
x
.
Show model answer
[QEx 7.9 Q8 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q265
#265
Mathematics → Integrals → Definite Integrals by Substitution
·
Hard
Add
The value of the integral
∫
1
/
3
1
(
x
−
x
3
)
1
/
3
x
4
d
x
\int_{1/3}^{1} \frac{(x-x^{3})^{1/3}}{x^{4}}\,dx
∫
1/3
1
x
4
(
x
−
x
3
)
1/3
d
x
is
A
6
B
0
C
3
D
4
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Show solution
[QEx 7.9 Q9 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q266
#266
Mathematics → Integrals → Definite Integrals by Substitution
·
Easy
Add
If
f
(
x
)
=
∫
0
x
t
sin
t
d
t
f(x) = \int_{0}^{x} t\sin t\,dt
f
(
x
)
=
∫
0
x
t
sin
t
d
t
, then
f
′
(
x
)
f'(x)
f
′
(
x
)
is
A
cos
x
+
x
sin
x
\cos x + x\sin x
cos
x
+
x
sin
x
B
x
sin
x
x\sin x
x
sin
x
C
x
cos
x
x\cos x
x
cos
x
D
sin
x
+
x
cos
x
\sin x + x\cos x
sin
x
+
x
cos
x
Tap an option to check your answer.
[QEx 7.9 Q10 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q267
#267
Mathematics → Integrals → Integration by Substitution
·
Moderate
Add
Find
∫
cos
6
x
1
+
sin
6
x
d
x
\int \cos 6x\, \sqrt{1+\sin 6x}\; dx
∫
cos
6
x
1
+
sin
6
x
d
x
.
Show model answer
[QMisc Eg.35 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q268
#268
Mathematics → Integrals → Integration by Substitution
·
Hard
Add
Find
∫
(
x
4
−
x
)
1
4
x
5
d
x
\int \frac{(x^4 - x)^{\frac{1}{4}}}{x^5}\, dx
∫
x
5
(
x
4
−
x
)
4
1
d
x
.
Show model answer
[QMisc Eg.36 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q269
#269
Mathematics → Integrals → Integration by Partial Fractions
·
Hard
Add
Find
∫
x
4
d
x
(
x
−
1
)
(
x
2
+
1
)
\int \frac{x^4\, dx}{(x-1)(x^2+1)}
∫
(
x
−
1
)
(
x
2
+
1
)
x
4
d
x
.
Show model answer
[QMisc Eg.37 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q270
#270
Mathematics → Integrals → Integration by Parts
·
Hard
Add
Find
∫
[
log
(
log
x
)
+
1
(
log
x
)
2
]
d
x
\int \left[\log(\log x) + \frac{1}{(\log x)^2}\right] dx
∫
[
lo
g
(
lo
g
x
)
+
(
l
o
g
x
)
2
1
]
d
x
.
Show model answer
[QMisc Eg.38 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q271
#271
Mathematics → Integrals → Integration by Substitution
·
Hard
Add
Find
∫
[
cot
x
+
tan
x
]
d
x
\int \left[\sqrt{\cot x} + \sqrt{\tan x}\right] dx
∫
[
cot
x
+
tan
x
]
d
x
.
Show model answer
[QMisc Eg.39 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q272
#272
Mathematics → Integrals → Integrals of Some Particular Functions
·
Hard
Add
Find
∫
sin
2
x
cos
2
x
d
x
9
−
cos
4
(
2
x
)
\int \frac{\sin 2x \cos 2x\, dx}{\sqrt{9 - \cos^4(2x)}}
∫
9
−
c
o
s
4
(
2
x
)
s
i
n
2
x
c
o
s
2
x
d
x
.
Show model answer
[QMisc Eg.40 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q273
#273
Mathematics → Integrals → Definite Integrals and the Fundamental Theorem
·
Hard
Add
Evaluate
∫
−
1
3
2
∣
x
sin
(
π
x
)
∣
d
x
\int_{-1}^{\frac{3}{2}} |x \sin(\pi x)|\, dx
∫
−
1
2
3
∣
x
sin
(
π
x
)
∣
d
x
.
Show model answer
[QMisc Eg.41 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q274
#274
Mathematics → Integrals → Properties of Definite Integrals
·
Hard
Add
Evaluate
∫
0
π
x
d
x
a
2
cos
2
x
+
b
2
sin
2
x
\int_{0}^{\pi} \frac{x\, dx}{a^2\cos^2 x + b^2\sin^2 x}
∫
0
π
a
2
c
o
s
2
x
+
b
2
s
i
n
2
x
x
d
x
.
Show model answer
[QMisc Eg.42 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q275
#275
Mathematics → Integrals → Integration by Partial Fractions
·
Moderate
Add
Integrate the function
1
x
−
x
3
\frac{1}{x - x^3}
x
−
x
3
1
.
Show model answer
[QMisc Q1 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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