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  • Explain Solution R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise Revision Exercise Question 116 Maths Textbook Solution.

Explain Solution R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise Revision Exercise  Question 116 Maths Textbook Solution.

Answers (1)

Answer:

2\left[x \sin ^{-1} x+\sqrt{1-x^{2}}\right]+C

Hint: to solve this statement we will suppose x as \sin \theta

Given: \int \cos ^{-1}(1-2 x)^{2} d x

Solution:

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

\operatorname{let} x=\sin \theta, d x=\cos \theta d \theta

I=\int \cos ^{-1}\left(1-2 \sin ^{2} \theta\right)^{2} \cos \theta d \theta

I=\int \cos ^{-1}(\cos 2 \theta) \cos \theta d \theta

I=\int 2 \theta \cos \theta d \theta

   \int u v d x=u \int v d x-\int\left[\frac{d u}{d v} \int v d x\right] d x

  I=2\left[\theta \sin \theta-\int \sin \theta d \theta\right]

I=2[\theta \sin \theta+\cos \theta]+c

I=2\left[x \sin ^{-1} x+\sqrt{1-x^{2}}\right]+C

 

 

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