NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11 are part of NCERT Solutions for Class 12 Maths. Here we have given NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11.
- 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 7.6
- Integrals Class 12 Ex 7.7
- Integrals Class 12 Ex 7.8
- Integrals Class 12 Ex 7.9
- Integrals Class 12 Ex 7.10
| Board | CBSE |
| Textbook | NCERT |
| Class | Class 12 |
| Subject | Maths |
| Chapter | Chapter 7 |
| Chapter Name | Integrals |
| Exercise | Ex 7.11 |
| Number of Questions Solved | 21 |
| Category | NCERT Solutions |
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11
By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19.
Ex 7.11 Class 12 Maths Question 1.
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { cos }^{ 2 }x\quad dx } =I\)
Solution:
\(I=\frac { 1 }{ 2 } \int _{ 0 }^{ \frac { \pi }{ 2 } }{ (1+cos2x)dx } =\frac { 1 }{ 2 } { \left[ x+\frac { sin2x }{ 2 } \right] }_{ 0 }^{ \frac { \pi }{ 2 } }\quad =\frac { \pi }{ 4 } \)
Ex 7.11 Class 12 Maths Question 2.
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { \sqrt { sinx } }{ \sqrt { sinx } +\sqrt { cosx } } dx } \)
Solution:
let I = \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { \sqrt { sinx } }{ \sqrt { sinx } +\sqrt { cosx } } dx } \)
Ex 7.11 Class 12 Maths Question 3.
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { { sin }^{ \frac { 3 }{ 2 } }xdx }{ { sin }^{ \frac { 3 }{ 2 } }x+{ cos }^{ \frac { 3 }{ 2 } }dx } dx } \)
Solution:
let I = \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { { sin }^{ \frac { 3 }{ 2 } }xdx }{ { sin }^{ \frac { 3 }{ 2 } }x+{ cos }^{ \frac { 3 }{ 2 } }dx } dx } \)
Ex 7.11 Class 12 Maths Question 4.
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { { cos }^{ 5 }xdx }{ { sin }^{ 5 }x+{ cos }^{ 5 }x } } \)
Solution:
let I = \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { { cos }^{ 5 }xdx }{ { sin }^{ 5 }x+{ cos }^{ 5 }x } } \)
Ex 7.11 Class 12 Maths Question 5.
\(\int _{ -5 }^{ 5 }{ \left| x+2 \right| dx=I } \)
Solution:
\(I=\int _{ -5 }^{ 5 }{ \left| x+2 \right| dx+\int _{ -2 }^{ 5 }{ \left| x+2 \right| dx } } \)
at x = – 5, x + 2 < 0; at x = – 2, x + 2 = 0; at x = 5, x + 2>0;x + 2<0, x + 2 = 0, x + 2>0
Ex 7.11 Class 12 Maths Question 6.
\(\int _{ 2 }^{ 8 }{ |x-5|dx } =I\)
Solution:
\(\int _{ 2 }^{ 8 }{ |x-5|dx } =I\)
Ex 7.11 Class 12 Maths Question 7.
\(\int _{ 0 }^{ 1 }{ x(1-x)^{ n }dx } =I\)
Solution:
\(\int _{ 0 }^{ 1 }{ x(1-x)^{ n }dx } =I\)
Ex 7.11 Class 12 Maths Question 8.
\(\int _{ 0 }^{ \frac { \pi }{ 4 } }{ log(1+tanx)dx } \)
Solution:
let I = \(\int _{ 0 }^{ \frac { \pi }{ 4 } }{ log(1+tanx)dx } \)
Ex 7.11 Class 12 Maths Question 9.
\(\int _{ 0 }^{ 2 }{ x\sqrt { 2-x } dx=I } \)
Solution:
let 2-x = t
⇒ – dx = dt
when x = 0, t = 2 and when x = 2,t = 0
\(\frac { 1 }{ 2 }\)
Ex 7.11 Class 12 Maths Question 10.
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \left( 2logsinx-logsin2x \right) dx=I } \)
Solution:
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \left( 2logsinx-logsin2x \right) dx=I } \)
Ex 7.11 Class 12 Maths Question 11.
\(\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ { sin }^{ 2 } } xdx\)
Solution:
Let f(x) = sin² x
f(-x) = sin² x = f(x)
∴ f(x) is an even function
\(\therefore \int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ { sin }^{ 2 } } xdx\quad =2\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \left[ \frac { 1-cos2x }{ 2 } \right] dx } \)
\(={ \left[ x-\frac { sin2x }{ x } \right] }_{ 0 }^{ \frac { \pi }{ 2 } }\therefore I=\frac { \pi }{ 2 } \)
Ex 7.11 Class 12 Maths Question 12.
\(\int _{ 0 }^{ \pi }{ \frac { xdx }{ 1+sinx } } \)
Solution:
let I = \(\int _{ 0 }^{ \pi }{ \frac { xdx }{ 1+sinx } } \) …(i)
Ex 7.11 Class 12 Maths Question 13.
\(\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ { sin }^{ 7 } } xdx\)
Solution:
Let f(x) = sin7 xdx
⇒ f(-x) = -sin7 x = -f(x)
⇒ f(x) is an odd function of x
⇒ \(\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ { sin }^{ 7 } } xdx=0\)
Ex 7.11 Class 12 Maths Question 14.
\(\int _{ 0 }^{ 2\pi }{ { cos }^{ 5 } } xdx\)
Solution:
let f(x) = cos5 x
⇒ f(2π – x) = cos5 x
Ex 7.11 Class 12 Maths Question 15.
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { sinx-cosx }{ 1+sinx\quad cosx } dx } \)
Solution:
let I = \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { sinx-cosx }{ 1+sinx\quad cosx } dx } \) …(i)
Ex 7.11 Class 12 Maths Question 16.
\(\int _{ 0 }^{ \pi }{ log(1+cosx)dx } \)
Solution:
let I = \(\int _{ 0 }^{ \pi }{ log(1+cosx)dx } \)
then I = \(\int _{ 0 }^{ \pi }{ log[1+cos(\pi -x)]dx } \)
Ex 7.11 Class 12 Maths Question 17.
\(\int _{ 0 }^{ a }{ \frac { \sqrt { x } }{ \sqrt { x } +\sqrt { a-x } } dx } \)
Solution:
let I = \(\int _{ 0 }^{ a }{ \frac { \sqrt { x } }{ \sqrt { x } +\sqrt { a-x } } dx } \) …(i)
Ex 7.11 Class 12 Maths Question 18.
\(\int _{ 0 }^{ 4 }{ \left| x-1 \right| dx=I } \)
Solution:
\(I=-\int _{ 0 }^{ 1 }{ (x-1)dx } +\int _{ 1 }^{ 4 }{ (x-1)dx } \)
\(=-{ \left[ \frac { { x }^{ 2 } }{ 2 } -x \right] }_{ 0 }^{ 1 }+{ \left[ \frac { { x }^{ 2 } }{ 2 } -x \right] }_{ 1 }^{ 4 }=5 \)
Ex 7.11 Class 12 Maths Question 19.
show that \(4\int _{ 0 }^{ a }{ f(x)g(x)dx } =2\int _{ 0 }^{ a }{ f(x)dx } \) if f and g are defined as f(x)=f(a-x) and g(x)+g(a-x)=4
Solution:
let I = \(\int _{ 0 }^{ a }{ f(x)g(x)dx } \)
Ex 7.11 Class 12 Maths Question 20.
The value of \(\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ \left( { x }^{ 3 }+xcosx+{ tan }^{ 5 }x+1 \right) dx } \) is
(a) 0
(b) 2
(c) π
(d) 1
Solution:
(c) let I = \(\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ \left( { x }^{ 3 }+xcosx+{ tan }^{ 5 }x+1 \right) dx } \) is
Ex 7.11 Class 12 Maths Question 21.
The value of \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ log\left[ \frac { 4+3sinx }{ 4+3sinx } \right] dx } \) is
(a) 2
(b) \(\frac { 3 }{ 4 }\)
(c) 0
(d) -2
Solution:
let I = \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ log\left[ \frac { 4+3sinx }{ 4+3sinx } \right] dx } \)
We hope the NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11 help you. If you have any query regarding NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11, drop a comment below and we will get back to you at the earliest.