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Exam: CBSE Class 12
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Q51
#51
Mathematics → Integrals → Properties of Definite Integrals
·
Easy
Add
By using the properties of definite integrals, evaluate the integral
∫
0
4
∣
x
−
1
∣
d
x
\int_{0}^{4} |x - 1|\,dx
∫
0
4
∣
x
−
1∣
d
x
.
Show model answer
[QEx 7.10 Q18 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q52
#52
Mathematics → Integrals → Properties of Definite Integrals
·
Hard
Add
Show that
∫
0
a
f
(
x
)
g
(
x
)
d
x
=
2
∫
0
a
f
(
x
)
d
x
\int_{0}^{a} f(x)g(x)\,dx = 2\int_{0}^{a} f(x)\,dx
∫
0
a
f
(
x
)
g
(
x
)
d
x
=
2
∫
0
a
f
(
x
)
d
x
, if
f
f
f
and
g
g
g
are defined as
f
(
x
)
=
f
(
a
−
x
)
f(x) = f(a - x)
f
(
x
)
=
f
(
a
−
x
)
and
g
(
x
)
+
g
(
a
−
x
)
=
4
g(x) + g(a - x) = 4
g
(
x
)
+
g
(
a
−
x
)
=
4
.
Show model answer
[QEx 7.10 Q19 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q53
#53
Mathematics → Integrals → Properties of Definite Integrals
·
Moderate
Add
The value of
∫
−
π
/
2
π
/
2
(
x
3
+
x
cos
x
+
tan
5
x
+
1
)
d
x
\int_{-\pi/2}^{\pi/2}(x^3 + x \cos x + \tan^5 x + 1)\,dx
∫
−
π
/2
π
/2
(
x
3
+
x
cos
x
+
tan
5
x
+
1
)
d
x
is
A
0
0
0
B
2
2
2
C
π
\pi
π
D
1
1
1
Tap an option to check your answer.
[QEx 7.10 Q20 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q54
#54
Mathematics → Integrals → Properties of Definite Integrals
·
Moderate
Add
The value of
∫
0
π
/
2
log
(
4
+
3
sin
x
4
+
3
cos
x
)
d
x
\int_{0}^{\pi/2} \log\left(\frac{4 + 3\sin x}{4 + 3\cos x}\right)\,dx
∫
0
π
/2
lo
g
(
4
+
3
c
o
s
x
4
+
3
s
i
n
x
)
d
x
is
A
2
2
2
B
3
4
\frac{3}{4}
4
3
C
0
0
0
D
−
2
-2
−
2
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[QEx 7.10 Q21 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Set · 4 questions
Integrate the following functions w.r.t.
x
x
x
:
Q55
#55
Mathematics → Integrals → Integration by Substitution
·
Moderate
Add
sin
m
x
\sin mx
sin
m
x
Show model answer
[Q7.2 Eg.5(i) · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q56
#56
Mathematics → Integrals → Integration by Substitution
·
Moderate
Add
2
x
sin
(
x
2
+
1
)
2x \sin (x^2 + 1)
2
x
sin
(
x
2
+
1
)
Show model answer
[Q7.2 Eg.5(ii) · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q57
#57
Mathematics → Integrals → Integration by Substitution
·
Hard
Add
tan
4
x
sec
2
x
x
\dfrac{\tan^4 \sqrt{x}\, \sec^2 \sqrt{x}}{\sqrt{x}}
x
tan
4
x
sec
2
x
Show model answer
[Q7.2 Eg.5(iii) · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q58
#58
Mathematics → Integrals → Integration by Substitution
·
Moderate
Add
sin
(
tan
−
1
x
)
1
+
x
2
\dfrac{\sin (\tan^{-1} x)}{1 + x^2}
1
+
x
2
sin
(
tan
−
1
x
)
Show model answer
[Q7.2 Eg.5(iv) · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
Report
Set · 3 questions
Find the following integrals:
Q59
#59
Mathematics → Integrals → Integration by Substitution
·
Hard
Add
∫
sin
3
x
cos
2
x
d
x
\int \sin^3 x \cos^2 x\,dx
∫
sin
3
x
cos
2
x
d
x
Show model answer
[Q7.2 Eg.6(i) · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q60
#60
Mathematics → Integrals → Integration by Substitution
·
Hard
Add
∫
sin
x
sin
(
x
+
a
)
d
x
\int \dfrac{\sin x}{\sin (x + a)}\,dx
∫
sin
(
x
+
a
)
sin
x
d
x
Show model answer
[Q7.2 Eg.6(ii) · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q61
#61
Mathematics → Integrals → Integration by Substitution
·
Hard
Add
∫
1
1
+
tan
x
d
x
\int \dfrac{1}{1 + \tan x}\,dx
∫
1
+
tan
x
1
d
x
Show model answer
[Q7.2 Eg.6(iii) · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q62
#62
Mathematics → Integrals → Integration by Substitution
·
Easy
Add
Integrate the function
2
x
1
+
x
2
\dfrac{2x}{1 + x^2}
1
+
x
2
2
x
.
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[QEx 7.2 Q1 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q63
#63
Mathematics → Integrals → Integration by Substitution
·
Easy
Add
Integrate the function
(
log
x
)
2
x
\dfrac{(\log x)^2}{x}
x
(
lo
g
x
)
2
.
Show model answer
[QEx 7.2 Q2 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
Report
Q64
#64
Mathematics → Integrals → Integration by Substitution
·
Moderate
Add
Integrate the function
1
x
+
x
log
x
\dfrac{1}{x + x \log x}
x
+
x
lo
g
x
1
.
Show model answer
[QEx 7.2 Q3 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q65
#65
Mathematics → Integrals → Integration by Substitution
·
Moderate
Add
Integrate the function
sin
x
sin
(
cos
x
)
\sin x \sin (\cos x)
sin
x
sin
(
cos
x
)
.
Show model answer
[QEx 7.2 Q4 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q66
#66
Mathematics → Integrals → Integration by Substitution
·
Moderate
Add
Integrate the function
sin
(
a
x
+
b
)
cos
(
a
x
+
b
)
\sin (ax + b) \cos (ax + b)
sin
(
a
x
+
b
)
cos
(
a
x
+
b
)
.
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[QEx 7.2 Q5 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q67
#67
Mathematics → Integrals → Integration by Substitution
·
Easy
Add
Integrate the function
a
x
+
b
\sqrt{ax + b}
a
x
+
b
.
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[QEx 7.2 Q6 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q68
#68
Mathematics → Integrals → Integration by Substitution
·
Moderate
Add
Integrate the function
x
x
+
2
x \sqrt{x + 2}
x
x
+
2
.
Show model answer
[QEx 7.2 Q7 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q69
#69
Mathematics → Integrals → Integration by Substitution
·
Moderate
Add
Integrate the function
x
1
+
2
x
2
x \sqrt{1 + 2x^2}
x
1
+
2
x
2
.
Show model answer
[QEx 7.2 Q8 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q70
#70
Mathematics → Integrals → Integration by Substitution
·
Moderate
Add
Integrate the function
(
4
x
+
2
)
x
2
+
x
+
1
(4x + 2) \sqrt{x^2 + x + 1}
(
4
x
+
2
)
x
2
+
x
+
1
.
Show model answer
[QEx 7.2 Q9 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q71
#71
Mathematics → Integrals → Integration by Substitution
·
Moderate
Add
Integrate the function
1
x
−
x
\dfrac{1}{x - \sqrt{x}}
x
−
x
1
.
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[QEx 7.2 Q10 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q72
#72
Mathematics → Integrals → Integration by Substitution
·
Moderate
Add
Integrate the function
x
x
+
4
,
x
>
0
\dfrac{x}{\sqrt{x + 4}},\ x > 0
x
+
4
x
,
x
>
0
.
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[QEx 7.2 Q11 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q73
#73
Mathematics → Integrals → Integration by Substitution
·
Hard
Add
Integrate the function
(
x
3
−
1
)
1
3
x
5
(x^3 - 1)^{\frac{1}{3}} x^5
(
x
3
−
1
)
3
1
x
5
.
Show model answer
[QEx 7.2 Q12 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q74
#74
Mathematics → Integrals → Integration by Substitution
·
Moderate
Add
Integrate the function
x
2
(
2
+
3
x
3
)
3
\dfrac{x^2}{(2 + 3x^3)^3}
(
2
+
3
x
3
)
3
x
2
.
Show model answer
[QEx 7.2 Q13 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q75
#75
Mathematics → Integrals → Integration by Substitution
·
Moderate
Add
Integrate the function
1
x
(
log
x
)
m
,
x
>
0
,
m
≠
1
\dfrac{1}{x (\log x)^m},\ x > 0,\ m \ne 1
x
(
lo
g
x
)
m
1
,
x
>
0
,
m
=
1
.
Show model answer
[QEx 7.2 Q14 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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