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Exam: CBSE Class 12
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Q176
#176
Mathematics → Integrals → Integration by Partial Fractions
·
Hard
Add
Integrate the rational function
1
x
4
−
1
\frac{1}{x^4-1}
x
4
−
1
1
.
Show model answer
[QEx 7.5 Q15 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q177
#177
Mathematics → Integrals → Integration by Partial Fractions
·
Hard
Add
Integrate the rational function
1
x
(
x
n
+
1
)
\frac{1}{x(x^n+1)}
x
(
x
n
+
1
)
1
. [Hint: multiply numerator and denominator by
x
n
−
1
x^{n-1}
x
n
−
1
and put
x
n
=
t
x^n=t
x
n
=
t
]
Show model answer
[QEx 7.5 Q16 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q178
#178
Mathematics → Integrals → Integration by Partial Fractions
·
Moderate
Add
Integrate the rational function
cos
x
(
1
−
sin
x
)
(
2
−
sin
x
)
\frac{\cos x}{(1-\sin x)(2-\sin x)}
(
1
−
s
i
n
x
)
(
2
−
s
i
n
x
)
c
o
s
x
. [Hint: Put
sin
x
=
t
\sin x=t
sin
x
=
t
]
Show model answer
[QEx 7.5 Q17 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q179
#179
Mathematics → Integrals → Integration by Partial Fractions
·
Hard
Add
Integrate the rational function
(
x
2
+
1
)
(
x
2
+
2
)
(
x
2
+
3
)
(
x
2
+
4
)
\frac{(x^2+1)(x^2+2)}{(x^2+3)(x^2+4)}
(
x
2
+
3
)
(
x
2
+
4
)
(
x
2
+
1
)
(
x
2
+
2
)
.
Show model answer
[QEx 7.5 Q18 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q180
#180
Mathematics → Integrals → Integration by Partial Fractions
·
Moderate
Add
Integrate the rational function
2
x
(
x
2
+
1
)
(
x
2
+
3
)
\frac{2x}{(x^2+1)(x^2+3)}
(
x
2
+
1
)
(
x
2
+
3
)
2
x
.
Show model answer
[QEx 7.5 Q19 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q181
#181
Mathematics → Integrals → Integration by Partial Fractions
·
Hard
Add
Integrate the rational function
1
x
(
x
4
−
1
)
\frac{1}{x(x^4-1)}
x
(
x
4
−
1
)
1
.
Show model answer
[QEx 7.5 Q20 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
Report
Q182
#182
Mathematics → Integrals → Integration by Partial Fractions
·
Moderate
Add
Integrate the rational function
1
e
x
−
1
\frac{1}{e^x-1}
e
x
−
1
1
. [Hint: Put
e
x
=
t
e^x=t
e
x
=
t
]
Show model answer
[QEx 7.5 Q21 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q183
#183
Mathematics → Integrals → Integration by Partial Fractions
·
Moderate
Add
∫
x
d
x
(
x
−
1
)
(
x
−
2
)
\int \frac{x\,dx}{(x-1)(x-2)}
∫
(
x
−
1
)
(
x
−
2
)
x
d
x
equals
A
log
∣
(
x
−
1
)
2
x
−
2
∣
+
C
\log\left|\frac{(x-1)^2}{x-2}\right| + C
lo
g
x
−
2
(
x
−
1
)
2
+
C
B
log
∣
(
x
−
2
)
2
x
−
1
∣
+
C
\log\left|\frac{(x-2)^2}{x-1}\right| + C
lo
g
x
−
1
(
x
−
2
)
2
+
C
C
log
∣
(
x
−
1
x
−
2
)
2
∣
+
C
\log\left|\left(\frac{x-1}{x-2}\right)^2\right| + C
lo
g
(
x
−
2
x
−
1
)
2
+
C
D
log
∣
(
x
−
1
)
(
x
−
2
)
∣
+
C
\log|(x-1)(x-2)| + C
lo
g
∣
(
x
−
1
)
(
x
−
2
)
∣
+
C
Tap an option to check your answer.
[QEx 7.5 Q22 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
Report
Q184
#184
Mathematics → Integrals → Integration by Partial Fractions
·
Moderate
Add
∫
d
x
x
(
x
2
+
1
)
\int \frac{dx}{x(x^2+1)}
∫
x
(
x
2
+
1
)
d
x
equals
A
log
∣
x
∣
−
1
2
log
(
x
2
+
1
)
+
C
\log|x| - \frac{1}{2}\log(x^2+1) + C
lo
g
∣
x
∣
−
2
1
lo
g
(
x
2
+
1
)
+
C
B
log
∣
x
∣
+
1
2
log
(
x
2
+
1
)
+
C
\log|x| + \frac{1}{2}\log(x^2+1) + C
lo
g
∣
x
∣
+
2
1
lo
g
(
x
2
+
1
)
+
C
C
−
log
∣
x
∣
+
1
2
log
(
x
2
+
1
)
+
C
-\log|x| + \frac{1}{2}\log(x^2+1) + C
−
lo
g
∣
x
∣
+
2
1
lo
g
(
x
2
+
1
)
+
C
D
1
2
log
∣
x
∣
+
log
(
x
2
+
1
)
+
C
\frac{1}{2}\log|x| + \log(x^2+1) + C
2
1
lo
g
∣
x
∣
+
lo
g
(
x
2
+
1
)
+
C
Tap an option to check your answer.
[QEx 7.5 Q23 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
Report
Q185
#185
Mathematics → Integrals → Integration by Parts
·
Easy
Add
Find
∫
x
cos
x
d
x
\int x \cos x \, dx
∫
x
cos
x
d
x
.
Show model answer
[Q7.6 Eg.17 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
Report
Q186
#186
Mathematics → Integrals → Integration by Parts
·
Moderate
Add
Find
∫
log
x
d
x
\int \log x \, dx
∫
lo
g
x
d
x
.
Show model answer
[Q7.6 Eg.18 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
Report
Q187
#187
Mathematics → Integrals → Integration by Parts
·
Easy
Add
Find
∫
x
e
x
d
x
\int x\, e^{x} \, dx
∫
x
e
x
d
x
.
Show model answer
[Q7.6 Eg.19 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
Report
Q188
#188
Mathematics → Integrals → Integration by Parts
·
Hard
Add
Find
∫
x
sin
−
1
x
1
−
x
2
d
x
\int \frac{x \sin^{-1} x}{\sqrt{1-x^{2}}} \, dx
∫
1
−
x
2
x
s
i
n
−
1
x
d
x
.
Show model answer
[Q7.6 Eg.20 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q189
#189
Mathematics → Integrals → Integration by Parts
·
Hard
Add
Find
∫
e
x
sin
x
d
x
\int e^{x} \sin x \, dx
∫
e
x
sin
x
d
x
.
Show model answer
[Q7.6 Eg.21 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Set · 2 questions
Find the following integrals.
Q190
#190
Mathematics → Integrals → Integration by Parts
·
Moderate
Add
Find
∫
e
x
(
tan
−
1
x
+
1
1
+
x
2
)
d
x
\int e^{x}\left(\tan^{-1} x + \frac{1}{1+x^{2}}\right) dx
∫
e
x
(
tan
−
1
x
+
1
+
x
2
1
)
d
x
.
Show model answer
[Q7.6 Eg.22(i) · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q191
#191
Mathematics → Integrals → Integration by Parts
·
Hard
Add
Find
∫
(
x
2
+
1
)
e
x
(
x
+
1
)
2
d
x
\int \frac{(x^{2}+1)\, e^{x}}{(x+1)^{2}} \, dx
∫
(
x
+
1
)
2
(
x
2
+
1
)
e
x
d
x
.
Show model answer
[Q7.6 Eg.22(ii) · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q192
#192
Mathematics → Integrals → Integration by Parts
·
Easy
Add
Integrate the function
x
sin
x
x \sin x
x
sin
x
.
Show model answer
[QEx 7.6 Q1 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q193
#193
Mathematics → Integrals → Integration by Parts
·
Easy
Add
Integrate the function
x
sin
3
x
x \sin 3x
x
sin
3
x
.
Show model answer
[QEx 7.6 Q2 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
Report
Q194
#194
Mathematics → Integrals → Integration by Parts
·
Moderate
Add
Integrate the function
x
2
e
x
x^{2} e^{x}
x
2
e
x
.
Show model answer
[QEx 7.6 Q3 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
Report
Q195
#195
Mathematics → Integrals → Integration by Parts
·
Easy
Add
Integrate the function
x
log
x
x \log x
x
lo
g
x
.
Show model answer
[QEx 7.6 Q4 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q196
#196
Mathematics → Integrals → Integration by Parts
·
Moderate
Add
Integrate the function
x
log
2
x
x \log 2x
x
lo
g
2
x
.
Show model answer
[QEx 7.6 Q5 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q197
#197
Mathematics → Integrals → Integration by Parts
·
Moderate
Add
Integrate the function
x
2
log
x
x^{2} \log x
x
2
lo
g
x
.
Show model answer
[QEx 7.6 Q6 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
Report
Q198
#198
Mathematics → Integrals → Integration by Parts
·
Moderate
Add
Integrate the function
x
sin
−
1
x
x \sin^{-1} x
x
sin
−
1
x
.
Show model answer
[QEx 7.6 Q7 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q199
#199
Mathematics → Integrals → Integration by Parts
·
Moderate
Add
Integrate the function
x
tan
−
1
x
x \tan^{-1} x
x
tan
−
1
x
.
Show model answer
[QEx 7.6 Q8 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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Q200
#200
Mathematics → Integrals → Integration by Parts
·
Moderate
Add
Integrate the function
x
cos
−
1
x
x \cos^{-1} x
x
cos
−
1
x
.
Show model answer
[QEx 7.6 Q9 · NCERT (CBSE Class 12) — Integrals (Chapter 7, NCERT Mathematics Part 2)]
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