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Exam: CBSE Class 12
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Q301
#301
Mathematics → Integrals → Definite Integrals by Substitution
·
Moderate
Add
Evaluate the definite integral
∫
π
6
π
3
sin
x
+
cos
x
sin
2
x
d
x
\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin x + \cos x}{\sqrt{\sin 2x}}\, dx
∫
6
π
3
π
s
i
n
2
x
s
i
n
x
+
c
o
s
x
d
x
.
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[QMisc Q27 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q302
#302
Mathematics → Integrals → Definite Integrals and the Fundamental Theorem
·
Moderate
Add
Evaluate the definite integral
∫
0
1
d
x
1
+
x
−
x
\int_{0}^{1} \frac{dx}{\sqrt{1 + x} - \sqrt{x}}
∫
0
1
1
+
x
−
x
d
x
.
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[QMisc Q28 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q303
#303
Mathematics → Integrals → Definite Integrals by Substitution
·
Hard
Add
Evaluate the definite integral
∫
0
π
4
sin
x
+
cos
x
9
+
16
sin
2
x
d
x
\int_{0}^{\frac{\pi}{4}} \frac{\sin x + \cos x}{9 + 16\sin 2x}\, dx
∫
0
4
π
9
+
16
s
i
n
2
x
s
i
n
x
+
c
o
s
x
d
x
.
Show model answer
[QMisc Q29 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q304
#304
Mathematics → Integrals → Definite Integrals by Substitution
·
Hard
Add
Evaluate the definite integral
∫
0
π
2
sin
2
x
tan
−
1
(
sin
x
)
d
x
\int_{0}^{\frac{\pi}{2}} \sin 2x \, \tan^{-1}(\sin x)\, dx
∫
0
2
π
sin
2
x
tan
−
1
(
sin
x
)
d
x
.
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[QMisc Q30 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q305
#305
Mathematics → Integrals → Definite Integrals and the Fundamental Theorem
·
Moderate
Add
Evaluate the definite integral
∫
1
4
[
∣
x
−
1
∣
+
∣
x
−
2
∣
+
∣
x
−
3
∣
]
d
x
\int_{1}^{4} \left[|x - 1| + |x - 2| + |x - 3|\right] dx
∫
1
4
[
∣
x
−
1∣
+
∣
x
−
2∣
+
∣
x
−
3∣
]
d
x
.
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[QMisc Q31 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q306
#306
Mathematics → Integrals → Integration by Partial Fractions
·
Moderate
Add
Prove that
∫
1
3
d
x
x
2
(
x
+
1
)
=
2
3
+
log
2
3
\int_{1}^{3} \frac{dx}{x^2 (x + 1)} = \frac{2}{3} + \log \frac{2}{3}
∫
1
3
x
2
(
x
+
1
)
d
x
=
3
2
+
lo
g
3
2
.
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[QMisc Q32 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q307
#307
Mathematics → Integrals → Integration by Parts
·
Easy
Add
Prove that
∫
0
1
x
e
x
d
x
=
1
\int_{0}^{1} x\, e^x\, dx = 1
∫
0
1
x
e
x
d
x
=
1
.
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[QMisc Q33 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q308
#308
Mathematics → Integrals → Properties of Definite Integrals
·
Moderate
Add
Prove that
∫
−
1
1
x
17
cos
4
x
d
x
=
0
\int_{-1}^{1} x^{17} \cos^4 x\, dx = 0
∫
−
1
1
x
17
cos
4
x
d
x
=
0
.
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[QMisc Q34 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q309
#309
Mathematics → Integrals → Integration using Trigonometric Identities
·
Moderate
Add
Prove that
∫
0
π
2
sin
3
x
d
x
=
2
3
\int_{0}^{\frac{\pi}{2}} \sin^3 x\, dx = \frac{2}{3}
∫
0
2
π
sin
3
x
d
x
=
3
2
.
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[QMisc Q35 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q310
#310
Mathematics → Integrals → Integration using Trigonometric Identities
·
Moderate
Add
Prove that
∫
0
π
4
2
tan
3
x
d
x
=
1
−
log
2
\int_{0}^{\frac{\pi}{4}} 2\tan^3 x\, dx = 1 - \log 2
∫
0
4
π
2
tan
3
x
d
x
=
1
−
lo
g
2
.
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[QMisc Q36 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q311
#311
Mathematics → Integrals → Integration by Parts
·
Moderate
Add
Prove that
∫
0
1
sin
−
1
x
d
x
=
π
2
−
1
\int_{0}^{1} \sin^{-1} x\, dx = \frac{\pi}{2} - 1
∫
0
1
sin
−
1
x
d
x
=
2
π
−
1
.
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[QMisc Q37 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q312
#312
Mathematics → Integrals → Integrals of Some Particular Functions
·
Moderate
Add
∫
d
x
e
x
+
e
−
x
\int \frac{dx}{e^x + e^{-x}}
∫
e
x
+
e
−
x
d
x
is equal to
A
tan
−
1
(
e
x
)
+
C
\tan^{-1}(e^x) + C
tan
−
1
(
e
x
)
+
C
B
tan
−
1
(
e
−
x
)
+
C
\tan^{-1}(e^{-x}) + C
tan
−
1
(
e
−
x
)
+
C
C
log
(
e
x
−
e
−
x
)
+
C
\log(e^x - e^{-x}) + C
lo
g
(
e
x
−
e
−
x
)
+
C
D
log
(
e
x
+
e
−
x
)
+
C
\log(e^x + e^{-x}) + C
lo
g
(
e
x
+
e
−
x
)
+
C
Tap an option to check your answer.
[QMisc Q38 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q313
#313
Mathematics → Integrals → Integration using Trigonometric Identities
·
Moderate
Add
∫
cos
2
x
(
sin
x
+
cos
x
)
2
d
x
\int \frac{\cos 2x}{(\sin x + \cos x)^2}\, dx
∫
(
s
i
n
x
+
c
o
s
x
)
2
c
o
s
2
x
d
x
is equal to
A
−
1
sin
x
+
cos
x
+
C
\frac{-1}{\sin x + \cos x} + C
s
i
n
x
+
c
o
s
x
−
1
+
C
B
log
∣
sin
x
+
cos
x
∣
+
C
\log|\sin x + \cos x| + C
lo
g
∣
sin
x
+
cos
x
∣
+
C
C
log
∣
sin
x
−
cos
x
∣
+
C
\log|\sin x - \cos x| + C
lo
g
∣
sin
x
−
cos
x
∣
+
C
D
1
(
sin
x
+
cos
x
)
2
\frac{1}{(\sin x + \cos x)^2}
(
s
i
n
x
+
c
o
s
x
)
2
1
Tap an option to check your answer.
[QMisc Q39 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q314
#314
Mathematics → Integrals → Properties of Definite Integrals
·
Moderate
Add
If
f
(
a
+
b
−
x
)
=
f
(
x
)
f(a + b - x) = f(x)
f
(
a
+
b
−
x
)
=
f
(
x
)
, then
∫
a
b
x
f
(
x
)
d
x
\int_{a}^{b} x\, f(x)\, dx
∫
a
b
x
f
(
x
)
d
x
is equal to
A
a
+
b
2
∫
a
b
f
(
b
−
x
)
d
x
\frac{a + b}{2}\int_{a}^{b} f(b - x)\, dx
2
a
+
b
∫
a
b
f
(
b
−
x
)
d
x
B
a
+
b
2
∫
a
b
f
(
b
+
x
)
d
x
\frac{a + b}{2}\int_{a}^{b} f(b + x)\, dx
2
a
+
b
∫
a
b
f
(
b
+
x
)
d
x
C
b
−
a
2
∫
a
b
f
(
x
)
d
x
\frac{b - a}{2}\int_{a}^{b} f(x)\, dx
2
b
−
a
∫
a
b
f
(
x
)
d
x
D
a
+
b
2
∫
a
b
f
(
x
)
d
x
\frac{a + b}{2}\int_{a}^{b} f(x)\, dx
2
a
+
b
∫
a
b
f
(
x
)
d
x
Tap an option to check your answer.
[QMisc Q40 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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