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Exam: Maharashtra HSC Class 12
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Q26
#26
Mathematics → Indefinite Integration → Elementary Integration and Standard Formulae
·
Moderate
Add
If
f
′
(
x
)
=
x
−
3
x
3
f'(x) = x - \dfrac{3}{x^3}
f
′
(
x
)
=
x
−
x
3
3
and
f
(
1
)
=
11
2
f(1) = \dfrac{11}{2}
f
(
1
)
=
2
11
, then find
f
(
x
)
f(x)
f
(
x
)
.
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[QEx 3.1 IV · Maharashtra State Board (Class 12) — Indefinite Integration (Balbharati textbook)]
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Set · 24 questions
Integrate the following functions w. r. t. x :
Q27
#27
Mathematics → Indefinite Integration → Integration by Substitution
·
Easy
Add
∫
(
log
x
)
n
x
d
x
\int \frac{(\log x)^n}{x}\, dx
∫
x
(
l
o
g
x
)
n
d
x
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[QEx 3.2(A) I (1) · Maharashtra State Board (Class 12) — Indefinite Integration (Balbharati textbook)]
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Q28
#28
Mathematics → Indefinite Integration → Integration by Substitution
·
Moderate
Add
∫
(
sin
−
1
x
)
3
2
1
−
x
2
d
x
\int \frac{(\sin^{-1} x)^{\frac{3}{2}}}{\sqrt{1 - x^2}}\, dx
∫
1
−
x
2
(
s
i
n
−
1
x
)
2
3
d
x
Show model answer
[QEx 3.2(A) I (2) · Maharashtra State Board (Class 12) — Indefinite Integration (Balbharati textbook)]
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Q29
#29
Mathematics → Indefinite Integration → Integration by Substitution
·
Moderate
Add
∫
1
+
x
x
⋅
sin
(
x
+
log
x
)
d
x
\int \frac{1 + x}{x \cdot \sin (x + \log x)}\, dx
∫
x
⋅
s
i
n
(
x
+
l
o
g
x
)
1
+
x
d
x
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[QEx 3.2(A) I (3) · Maharashtra State Board (Class 12) — Indefinite Integration (Balbharati textbook)]
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Q30
#30
Mathematics → Indefinite Integration → Integration by Substitution
·
Moderate
Add
∫
x
⋅
sec
2
(
x
2
)
tan
3
(
x
2
)
d
x
\int \frac{x \cdot \sec^2 (x^2)}{\sqrt{\tan^3 (x^2)}}\, dx
∫
t
a
n
3
(
x
2
)
x
⋅
s
e
c
2
(
x
2
)
d
x
Show model answer
[QEx 3.2(A) I (4) · Maharashtra State Board (Class 12) — Indefinite Integration (Balbharati textbook)]
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Q31
#31
Mathematics → Indefinite Integration → Integration by Substitution
·
Easy
Add
∫
e
3
x
e
3
x
+
1
d
x
\int \frac{e^{3x}}{e^{3x} + 1}\, dx
∫
e
3
x
+
1
e
3
x
d
x
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[QEx 3.2(A) I (5) · Maharashtra State Board (Class 12) — Indefinite Integration (Balbharati textbook)]
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Q32
#32
Mathematics → Indefinite Integration → Integration by Substitution
·
Moderate
Add
∫
(
x
2
+
2
)
(
x
2
+
1
)
⋅
a
x
+
tan
−
1
x
d
x
\int \frac{(x^2 + 2)}{(x^2 + 1)} \cdot a^{x + \tan^{-1} x}\, dx
∫
(
x
2
+
1
)
(
x
2
+
2
)
⋅
a
x
+
t
a
n
−
1
x
d
x
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[QEx 3.2(A) I (6) · Maharashtra State Board (Class 12) — Indefinite Integration (Balbharati textbook)]
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Q33
#33
Mathematics → Indefinite Integration → Integration by Substitution
·
Moderate
Add
∫
e
x
⋅
log
(
sin
e
x
)
tan
(
e
x
)
d
x
\int \frac{e^x \cdot \log (\sin e^x)}{\tan (e^x)}\, dx
∫
t
a
n
(
e
x
)
e
x
⋅
l
o
g
(
s
i
n
e
x
)
d
x
Show model answer
[QEx 3.2(A) I (7) · Maharashtra State Board (Class 12) — Indefinite Integration (Balbharati textbook)]
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Q34
#34
Mathematics → Indefinite Integration → Integration by Substitution
·
Moderate
Add
∫
e
2
x
+
1
e
2
x
−
1
d
x
\int \frac{e^{2x} + 1}{e^{2x} - 1}\, dx
∫
e
2
x
−
1
e
2
x
+
1
d
x
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[QEx 3.2(A) I (8) · Maharashtra State Board (Class 12) — Indefinite Integration (Balbharati textbook)]
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Q35
#35
Mathematics → Indefinite Integration → Integrals of Trigonometric Functions
·
Moderate
Add
∫
sin
4
x
⋅
cos
3
x
d
x
\int \sin^4 x \cdot \cos^3 x\, dx
∫
sin
4
x
⋅
cos
3
x
d
x
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[QEx 3.2(A) I (9) · Maharashtra State Board (Class 12) — Indefinite Integration (Balbharati textbook)]
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Q36
#36
Mathematics → Indefinite Integration → Integration by Substitution
·
Hard
Add
∫
1
4
x
+
5
x
−
11
d
x
\int \frac{1}{4x + 5x^{-11}}\, dx
∫
4
x
+
5
x
−
11
1
d
x
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[QEx 3.2(A) I (10) · Maharashtra State Board (Class 12) — Indefinite Integration (Balbharati textbook)]
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Q37
#37
Mathematics → Indefinite Integration → Integration by Substitution
·
Easy
Add
∫
x
9
⋅
sec
2
(
x
10
)
d
x
\int x^9 \cdot \sec^2 (x^{10})\, dx
∫
x
9
⋅
sec
2
(
x
10
)
d
x
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[QEx 3.2(A) I (11) · Maharashtra State Board (Class 12) — Indefinite Integration (Balbharati textbook)]
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Q38
#38
Mathematics → Indefinite Integration → Integration by Substitution
·
Moderate
Add
∫
e
3
log
x
⋅
(
x
4
+
1
)
−
1
d
x
\int e^{3 \log x} \cdot (x^4 + 1)^{-1}\, dx
∫
e
3
l
o
g
x
⋅
(
x
4
+
1
)
−
1
d
x
Show model answer
[QEx 3.2(A) I (12) · Maharashtra State Board (Class 12) — Indefinite Integration (Balbharati textbook)]
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Q39
#39
Mathematics → Indefinite Integration → Integrals of Trigonometric Functions
·
Moderate
Add
∫
tan
x
sin
x
⋅
cos
x
d
x
\int \frac{\sqrt{\tan x}}{\sin x \cdot \cos x}\, dx
∫
s
i
n
x
⋅
c
o
s
x
t
a
n
x
d
x
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[QEx 3.2(A) I (13) · Maharashtra State Board (Class 12) — Indefinite Integration (Balbharati textbook)]
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Q40
#40
Mathematics → Indefinite Integration → Integration by Substitution
·
Hard
Add
∫
(
x
−
1
)
2
(
x
2
+
1
)
2
d
x
\int \frac{(x - 1)^2}{(x^2 + 1)^2}\, dx
∫
(
x
2
+
1
)
2
(
x
−
1
)
2
d
x
Show model answer
[QEx 3.2(A) I (14) · Maharashtra State Board (Class 12) — Indefinite Integration (Balbharati textbook)]
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Q41
#41
Mathematics → Indefinite Integration → Integrals of Trigonometric Functions
·
Moderate
Add
∫
2
sin
x
⋅
cos
x
3
cos
2
x
+
4
sin
2
x
d
x
\int \frac{2 \sin x \cdot \cos x}{3 \cos^2 x + 4 \sin^2 x}\, dx
∫
3
c
o
s
2
x
+
4
s
i
n
2
x
2
s
i
n
x
⋅
c
o
s
x
d
x
Show model answer
[QEx 3.2(A) I (15) · Maharashtra State Board (Class 12) — Indefinite Integration (Balbharati textbook)]
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Q42
#42
Mathematics → Indefinite Integration → Integration by Substitution
·
Moderate
Add
∫
1
x
+
x
3
d
x
\int \frac{1}{\sqrt{x} + \sqrt{x^3}}\, dx
∫
x
+
x
3
1
d
x
Show model answer
[QEx 3.2(A) I (16) · Maharashtra State Board (Class 12) — Indefinite Integration (Balbharati textbook)]
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Q43
#43
Mathematics → Indefinite Integration → Integration by Substitution
·
Moderate
Add
∫
10
x
9
+
10
x
⋅
log
10
10
x
+
x
10
d
x
\int \frac{10 x^9 + 10^x \cdot \log 10}{10^x + x^{10}}\, dx
∫
1
0
x
+
x
10
10
x
9
+
1
0
x
⋅
l
o
g
10
d
x
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[QEx 3.2(A) I (17) · Maharashtra State Board (Class 12) — Indefinite Integration (Balbharati textbook)]
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Q44
#44
Mathematics → Indefinite Integration → Integration by Substitution
·
Moderate
Add
∫
x
n
−
1
1
+
4
x
n
d
x
\int \frac{x^{n-1}}{\sqrt{1 + 4x^n}}\, dx
∫
1
+
4
x
n
x
n
−
1
d
x
Show model answer
[QEx 3.2(A) I (18) · Maharashtra State Board (Class 12) — Indefinite Integration (Balbharati textbook)]
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Q45
#45
Mathematics → Indefinite Integration → Integration by Substitution
·
Moderate
Add
∫
(
2
x
+
1
)
x
+
2
d
x
\int (2x + 1)\sqrt{x + 2}\, dx
∫
(
2
x
+
1
)
x
+
2
d
x
Show model answer
[QEx 3.2(A) I (19) · Maharashtra State Board (Class 12) — Indefinite Integration (Balbharati textbook)]
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Q46
#46
Mathematics → Indefinite Integration → Integration by Substitution
·
Hard
Add
∫
x
5
⋅
a
2
+
x
2
d
x
\int x^5 \cdot \sqrt{a^2 + x^2}\, dx
∫
x
5
⋅
a
2
+
x
2
d
x
Show model answer
[QEx 3.2(A) I (20) · Maharashtra State Board (Class 12) — Indefinite Integration (Balbharati textbook)]
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Q47
#47
Mathematics → Indefinite Integration → Integration by Substitution
·
Moderate
Add
∫
(
5
−
3
x
)
(
2
−
3
x
)
−
1
2
d
x
\int (5 - 3x)(2 - 3x)^{-\frac{1}{2}}\, dx
∫
(
5
−
3
x
)
(
2
−
3
x
)
−
2
1
d
x
Show model answer
[QEx 3.2(A) I (21) · Maharashtra State Board (Class 12) — Indefinite Integration (Balbharati textbook)]
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Q48
#48
Mathematics → Indefinite Integration → Integration by Substitution
·
Hard
Add
∫
7
+
4
x
+
5
x
2
(
2
x
+
3
)
3
2
d
x
\int \frac{7 + 4x + 5x^2}{(2x + 3)^{\frac{3}{2}}}\, dx
∫
(
2
x
+
3
)
2
3
7
+
4
x
+
5
x
2
d
x
Show model answer
[QEx 3.2(A) I (22) · Maharashtra State Board (Class 12) — Indefinite Integration (Balbharati textbook)]
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Q49
#49
Mathematics → Indefinite Integration → Integration by Substitution
·
Moderate
Add
∫
x
2
9
−
x
6
d
x
\int \frac{x^2}{\sqrt{9 - x^6}}\, dx
∫
9
−
x
6
x
2
d
x
Show model answer
[QEx 3.2(A) I (23) · Maharashtra State Board (Class 12) — Indefinite Integration (Balbharati textbook)]
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Q50
#50
Mathematics → Indefinite Integration → Integration by Substitution
·
Hard
Add
∫
1
x
(
x
3
−
1
)
d
x
\int \frac{1}{x (x^3 - 1)}\, dx
∫
x
(
x
3
−
1
)
1
d
x
Show model answer
[QEx 3.2(A) I (24) · Maharashtra State Board (Class 12) — Indefinite Integration (Balbharati textbook)]
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