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Please Solve R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise  Revision Exercise Question 38 Maths Textbbok Solution.

Answers (1)

Answer:

\frac{2}{3} \cos ^{3} x-\frac{1}{5} \cos ^{5} x-\cos x+c

Given:

\int \sin ^{5} x d x

Hint:

To solve the statement we have to convert sin x into cos x.

Solution: 

\int \sin x(\sin x)^{4} d x

   \int \sin x\left(\sin ^{2} x\right)^{2} d x                            \left[\because \sin ^{2} \theta+\cos ^{2} \theta=1, \sin ^{2} \theta=1-\cos ^{2} \theta\right]

\int \sin x\left(1-\cos ^{2} x\right)^{2} d x

 Let cosx= t

-\sin x d x=d t

\sin x d x=-d t

I=-\int\left(1-t^{2}\right)^{2} d t

I=-\int\left(1+t^{4}-2 t^{2}\right) d t

I=\int\left(2 t^{2}-t^{4}-1\right) d t \quad\left[\because \int x^{n} d x=\frac{x^{n+1}}{n+1}\right]

I=\frac{2 t^{3}}{3}-\frac{t^{5}}{5}-t+c

I=\frac{2}{3} \cos ^{3} x-\frac{1}{5} \cos ^{5} x-\cos x+c

 

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