NCERT Solutions for Class 12 Maths Exercise 7.3 hapter 7 Integrals – FREE PDF Download

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NCERT Solutions for Class 12 Maths Chapter 7 Integrals (Ex 7.3) Exercise 7.3

Find the integrals of the following functions in Exercises 1 to 9.

Ans.

Using $\sin^{2} θ = \frac{1 - \cos 2 θ}{2}$

Using $\int \cos (a x + b) d x = \frac{\sin (a x + b)}{a} + c$

$= \frac{1}{2} [x - \frac{\sin (4 x + 10)}{4}] + c$

2.

Ans.

= $= \frac{1}{2} \int {\sin (4 x + 3 x) - \sin (4 x - 3 x)} d x$ Using 2sinB cosA=sin(A+B)-sin(A-B)

3.

Ans.

Using 2cosA cosB=cos(A+B)+cos(A-B)

and $2 \cos^{2} θ = 1 + \cos 2 θ$

4.

Ans.

Another Method

$= \int {Sin}^{2} (2 x + 1) Sin (2 x + 1) d x$

$= \int (1 - \cos^{2} (2 x + 1)) Sin (2 x + 1) d x$

Using $\sin^{2} θ = 1 - \cos^{2} θ$

let $\cos (2 x + 1) = t$

therefore $- 2 \sin (2 x + 1) d x = d t$

thus $\sin (2 x + 1) d x = - \frac{1}{2} d t$

$= - \frac{1}{2} \int (1 - t^{2}) d t$

$= - \frac{1}{2} (\int 1 d t - \int t^{2} d t)$

$= - \frac{1}{2} (t - \frac{t^{3}}{3}) + c$

$= - \frac{1}{2} Cos (2 x + 1) + \frac{1}{6} {Cos}^{3} (2 x + 1) + c$ Ans

5.

Ans.

Another method let I=

$I = \int (1 - \cos^{2} x) \cos^{3} x \sin x d x$

$= \int (\cos^{3} x - \cos^{5} x) \sin x d x$

let cosx=t

-sinx dx=dt

sinx dx = -dt

$I = - \int (t^{3} - t^{5}) d t$

$I = \int (t^{5} - t^{3}) d t$

$= \frac{t^{6}}{6} - \frac{t^{4}}{4} + c$

$= \frac{\cos^{6}}{6} - \frac{\cos^{4}}{4} + c$

6.

Ans.

= Using 2sin A sin B=cos (A-B) – cos ( A+B)

$= \frac{1}{2} \int (\cos x \sin x - \cos 5 x \sin x) d x)$

$= \frac{1}{4} \int (2 \cos x \sin x - 2 \cos 5 x \sin x) d x)$

= $= \frac{1}{4} \int (2 \sin x \cos x - {\sin (5 x + x) - \sin (5 x - x)}) d x$

using 2cosA sinB=sin(A+B)-sin(A-B)

as $\begin{aligned} \int \sin (a x + b) d x = \frac{- \cos (a x + b)}{a} + c \end{aligned}$

= Ans

7.

Ans.

= Using 2sinA sinB= cos(A-B)-cos(A+B)

$= \frac{1}{2} \int (\cos (4 x - 8 x) - \cos (4 x + 8 x)) d x$

As cos(-x)=cosx

= Ans. As $\int \cos (a x + b) d x = \frac{\sin (a x + b)}{a} + c$

8.

Ans.

As $\begin{aligned} 2 \sin^{2} \frac{θ}{2} = 1 - \cos θ \\ 2 \cos^{2} \frac{θ}{2} = 1 + \cos θ \end{aligned}$

As $\int \sec^{2} (a x + b) d x = \frac{\tan (a x + b)}{a} + c$

9.

Ans.

As $2 \cos^{2} \frac{θ}{2} = 1 + \cos θ$

as $\int \sec^{2} (a x + b) d x = \frac{\tan (a x + b)}{a} + c$

Find the integrals of the following functions in Exercises 10 to 18.

10.

Ans.

as $2 \sin^{2} θ = 1 - \cos 2 θ$

as $2 \cos^{2} 2 θ = 1 + \cos 4 θ$

As $\int \cos (a x + b) d x = \frac{\sin (a x + b)}{a} + c$

= Ans

11.

Ans.

As $2 \cos^{2} 2 θ = 1 + \cos 4 θ$

as $2 \cos^{2} 4 θ = 1 + \cos 8 θ$

using $\int \cos (a x + b) d x = \frac{\sin (a x + b)}{a} + c$

12.

Ans.

13.

Ans.

using $\cos 2 θ = 2 \cos^{2} θ - 1$

14.

Ans. Let I =

using identity $1 = \sin^{2} θ + \cos^{2} θ$

= …..(i)

Putting

From eq. (i), I =

15.

Ans. Let I =

Putting

From eq. (i),

I =

16.

Ans.

= ……….(i)

Putting

From eq. (i),

17.

Ans.

18.

Ans.

using $\cos 2 θ = 1 - 2 \sin^{2} θ$

Integrate the following functions in Exercises 19 to 22.

19.

Ans. Let I =

……….(i)

Dividing numerator and denominator by $\cos^{4} θ$

$= \int \frac{1 / \cos^{4} x d x}{\sin x / \cos x}$

$I = \int \frac{\sec^{4} x d x}{\tan x}$

$I = \int \frac{(1 + \tan^{2} x) \sec^{2} x d x}{\tan x}$ ……….(ii)

Putting

From eq. (ii),

I =

20.

Ans. Let I =

= ……….(i)

Putting

From eq. (i),

I =

21.

Ans.

22.

Ans. Let I =

……….(i)

Multiplying and dividing by

as ,

I =

using sin(A-B)=sinAcosB-cosAsinB

Choose the correct answer in Exercise 23 and 24.

23. is equal to:

(A)

(B)

(C)

(D)

Ans.

Therefore, option (A) is correct.

24. is equal to:

(A)

(B)

(C)

(D)

Ans. Let I =

……….(i)

Putting

From eq. (i),

I =

Therefore, option (B) is correct.

NCERT Solutions class 12 Maths Exercise 7.3 (Ex 7.3) Chapter 7 Integrals

NCERT Solutions for Class 12 Maths Exercise 7.3 hapter 7 Integrals – FREE PDF Download

NCERT Solutions for Class 12 Maths Chapter 7 Integrals (Ex 7.3) Exercise 7.3

2.

3.

4.

5.

6.

7.

8.

9.

Find the integrals of the following functions in Exercises 10 to 18.

11.

12.

13.

14.

15.

16.

17.

18.

Integrate the following functions in Exercises 19 to 22.

20.

21.

22.

Choose the correct answer in Exercise 23 and 24.

24. is equal to: