NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 are part of NCERT Solutions for Class 12 Maths. Here we have given NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6.
- Integrals Class 12 Ex 7.1
- Integrals Class 12 Ex 7.2
- Integrals Class 12 Ex 7.3
- Integrals Class 12 Ex 7.4
- Integrals Class 12 Ex 7.5
- Integrals Class 12 Ex 7.7
- Integrals Class 12 Ex 7.8
- Integrals Class 12 Ex 7.9
- Integrals Class 12 Ex 7.10
- Integrals Class 12 Ex 7.11
| Board | CBSE |
| Textbook | NCERT |
| Class | Class 12 |
| Subject | Maths |
| Chapter | Chapter 7 |
| Chapter Name | Integrals |
| Exercise | Ex 7.6 |
| Number of Questions Solved | 24 |
| Category | NCERT Solutions |
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6
Integrate the functions in Exercises 1 to 22.
Ex 7.6 Class 12 Maths Question 1.
x sinx
Solution:
By part integration
∫x sinx dx = x(-cosx) – ∫1(-cosx)dx
=-x cosx + ∫cosxdx
=-x cosx + sinx + c
Ex 7.6 Class 12 Maths Question 2.
x sin3x
Solution:
∫x sin3x dx = \(x\left( -\frac { cos3x }{ 3 } \right) -\int { 1 } .\left( \frac { -cos3x }{ 3 } \right) dx\)
\(=-\frac { 1 }{ 3 } x\quad cos3x+\frac { 1 }{ 9 } sin3x+c\)
Ex 7.6 Class 12 Maths Question 3.
\({ x }^{ 2 }{ e }^{ x }\)
Solution:
\(\int { { x }^{ 2 }{ e }^{ x } } dx={ x }^{ 2 }{ e }^{ x }-2{ x }{ e }^{ x }+2{ e }^{ x }+c\)
\(={ e }^{ x }\left( { x }^{ 2 }-2x+2 \right) +c\)
Ex 7.6 Class 12 Maths Question 4.
x logx
Solution:
\(\int { xlogx\quad dx } =logx\int { xdx } -\int { \left[ \frac { d }{ dx } (logx)\int { xdx } \right] dx } \)
\(=\frac { { x }^{ 2 } }{ 2 } logx-\frac { 1 }{ 2 } \int { x\quad dx } =\frac { { x }^{ 2 } }{ 2 } logx-\frac { 1 }{ 4 } { x }^{ 2 }+c \)
Ex 7.6 Class 12 Maths Question 5.
x log2x
Solution:
\(\int { x\quad log2xdx } =(log2x)\frac { { x }^{ 2 } }{ 2 } -\int { \frac { 1 }{ 2x } } .2\left( \frac { { x }^{ 2 } }{ 2 } \right) dx\)
\(=\frac { { x }^{ 2 } }{ 2 } log|2x|-\frac { 1 }{ 2 } \int { xdx } =\frac { { x }^{ 2 } }{ 2 } log|2x|-\frac { { x }^{ 2 } }{ 4 } +c\)
Ex 7.6 Class 12 Maths Question 6.
\({ x }^{ 2 }logx\)
Solution:
\(\int { { x }^{ 2 }logxdx } =log|x|\left( \frac { { x }^{ 3 } }{ 3 } \right) -\int { \frac { 1 }{ x } } \left( \frac { { x }^{ 3 } }{ 3 } \right) dx\)
\(=\frac { { x }^{ 3 } }{ 3 } log|x|-\frac { 1 }{ 3 } \int { { x }^{ 2 }dx } =\frac { { x }^{ 3 } }{ 3 } log|x|-\frac { { x }^{ 3 } }{ 9 } +c\)
Ex 7.6 Class 12 Maths Question 7.
\(x\quad { sin }^{ -1 }x\)
Solution:
\(I=x\quad { sin }^{ -1 }x.\left( \frac { { x }^{ 2 } }{ 2 } \right) -\int { \frac { 1 }{ \sqrt { 1-{ x }^{ 2 } } } } .\frac { { x }^{ 2 } }{ 2 } dx\)
Ex 7.6 Class 12 Maths Question 8.
\(x\quad { tan }^{ -1 }x\)
Solution:
\(I=x\quad { tan}^{ -1 }x.\left( \frac { { x }^{ 2 } }{ 2 } \right) -\int { \frac { 1 }{ \sqrt { 1+{ x }^{ 2 } } } } .\frac { { x }^{ 2 } }{ 2 } dx\)
\(=\frac { { x }^{ 2 } }{ 2 } { tan }^{ -1 }x-\frac { 1 }{ 2 } \int { \left( 1-\frac { 1 }{ 1+{ x }^{ 2 } } \right) dx } \)
\(=\frac { { x }^{ 2 } }{ 2 } { tan }^{ -1 }x-\frac { 1 }{ 2 } x+\frac { 1 }{ 2 } { tan }^{ -1 }x+c\)
Ex 7.6 Class 12 Maths Question 9.
\(x\quad { cos }^{ -1 }x\)
Solution:
let I = \(\int { x } { cos }^{ -1 }xdx=\int { { cos }^{ -1 }x } .xdx\)
Ex 7.6 Class 12 Maths Question 10.
\({ (sin }^{ -1 }{ x })^{ 2 }\)
Solution:
\(put\quad { sin }^{ -1 }x=\theta \Rightarrow x=sin\theta \Rightarrow dx=cos\theta d\theta \)
Ex 7.6 Class 12 Maths Question 11.
\(\frac { x\quad { cos }^{ -1 }x }{ \sqrt { 1-{ x }^{ 2 } } } \)
Solution:
\(put\quad { cos }^{ -1 }x=t\quad so\quad that\frac { x\quad { cos }^{ -1 }x }{ \sqrt { 1-{ x }^{ 2 } } } dx=dt\)
Ex 7.6 Class 12 Maths Question 12.
x sec²x
Solution:
∫x sec²x dx =x(tanx)-∫1.tanx dx
= x tanx+log cosx+c
Ex 7.6 Class 12 Maths Question 13.
\({ ta }n^{ -1 }x\)
Solution:
\(\int { { tan }^{ -1 }xdx } =x{ tan }^{ -1 }x-\frac { 1 }{ 2 } \int { \frac { 2x }{ 1+{ x }^{ 2 } } dx } \)
\(=x{ tan }^{ -1 }x-\frac { 1 }{ 2 } log|1+{ x }^{ 2 }|+c \)
Ex 7.6 Class 12 Maths Question 14.
x(logx)²
Solution:
∫x(logx)² dx
\(=\frac { { x }^{ 2 } }{ 2 } { (logx) }^{ 2 }-\left[ (logx)\frac { { x }^{ 2 } }{ 2 } -\int { \frac { 1 }{ x } \frac { { x }^{ 2 } }{ 2 } dx } \right] \)
\(=\frac { { x }^{ 2 } }{ 2 } { (logx) }^{ 2 }-\frac { { x }^{ 2 } }{ 2 } logx+\frac { 1 }{ 4 } { x }^{ 2 }+c\)
Ex 7.6 Class 12 Maths Question 15.
(x²+1)logx
Solution:
∫(x²+1)logx dx
\(=logx\left( \frac { { x }^{ 3 } }{ 3 } +x \right) -\int { \frac { 1 }{ x } \left( \frac { { x }^{ 3 } }{ 3 } +x \right) dx } \)
\(=\left( \frac { { x }^{ 3 } }{ 3 } +x \right) logx-\frac { { x }^{ 3 } }{ 9 } -x+c\)
Ex 7.6 Class 12 Maths Question 16.
\({ e }^{ x }(sinx+cosx)\)
Solution:
\(put\quad { e }^{ x }sinx=t\Rightarrow { e }^{ x }(sinx+cosx)dx=dt\)
\(\therefore \int { { e }^{ x }(sinx+cosx)dx } =\int { dt } =t+c\)
\(={ e }^{ x }sinx+c\)
Ex 7.6 Class 12 Maths Question 17.
\(\frac { { xe }^{ x } }{ { (1+x) }^{ 2 } } \)
Solution:
\(\int { \frac { { xe }^{ x } }{ { (1+x) }^{ 2 } } } \)
Ex 7.6 Class 12 Maths Question 18.
\(\frac { { e }^{ x }(1+sinx) }{ 1+cosx } \)
Solution:
\(I=\int { { e }^{ x } } \left[ \frac { 1+2sin\frac { x }{ 2 } cos\frac { x }{ 2 } }{ 2{ cos }^{ 2 }\frac { x }{ 2 } } \right] dx\)
Ex 7.6 Class 12 Maths Question 19.
\({ e }^{ x }\left( \frac { 1 }{ x } -\frac { 1 }{ { x }^{ 2 } } \right) \)
Solution:
put \(\frac { { e }^{ x } }{ x } =t\Rightarrow { e }^{ x }\left( \frac { 1 }{ x } -\frac { 1 }{ { x }^{ 2 } } \right) dx=dt\)
\(\therefore I=\int { dt } =t+c=\frac { { e }^{ x } }{ x } +c\)
Ex 7.6 Class 12 Maths Question 20.
\(\frac { { (x-2)e }^{ x } }{ { (x-1) }^{ 3 } } \)
Solution:
\(I=\int { { e }^{ x }\left[ \frac { 1 }{ { (x-1) }^{ 2 } } -\frac { 2 }{ { (x-1) }^{ 3 } } \right] dx } \)
Ex 7.6 Class 12 Maths Question 21.
\({ e }^{ 2x }sinx\)
Solution:
let \(I=\int { { e }^{ 2x }sinx } \)
\(={ e }^{ 2x }(-cosx)-\int { 2{ e }^{ 2x }(-cosx)dx } \)
Ex 7.6 Class 12 Maths Question 22.
\({ sin }^{ -1 }\left( \frac { 2x }{ 1+{ x }^{ 2 } } \right) \)
Solution:
Put x = tan t
so that dx = sec² t dt
Choose the correct answer in exercise 23 and 24
Ex 7.6 Class 12 Maths Question 23.
\(\int { { x }^{ 2 }{ e }^{ { x }^{ 3 } } } dx\quad equals\)
(a) \(\frac { 1 }{ 3 } { e }^{ { x }^{ 3 } }+c\)
(b) \(\frac { 1 }{ 3 } +{ e }^{ { x }^{ 2 } }+c\)
(c) \(\frac { 1 }{ 2 } { e }^{ { x }^{ 3 } }+c\)
(d) \(\frac { 1 }{ 2 } { e }^{ { x }^{ 2 } }+c\)
Solution:
(a) let x³ = t
⇒3x² dx = dt
\(\therefore \int { { x }^{ 2 }{ e }^{ { x }^{ 3 } }dx } =\frac { 1 }{ 3 } \int { { e }^{ t }dt } =\frac { 1 }{ 3 } { e }^{ t }+c=\frac { 1 }{ 3 } { e }^{ { x }^{ 3 } }+c\)
Ex 7.6 Class 12 Maths Question 24.
\(\int { { e }^{ x }secx(1+tanx) } dx\quad equals\)
(a) \({ e }^{ x }cosx+c\)
(b) \({ e }^{ x }secx+c\)
(c) \({ e }^{ x }sinx+c\)
(d) \({ e }^{ x }tanx+c\)
Solution:
(b) \(\int { { e }^{ x }(secx+secx\quad tanx)dx } ={ e }^{ x }secx+c\)
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