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  • Explain Solution R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise Revision Exercise Question 113 Maths Textbook Solution.

Explain Solution R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise Revision Exercise  Question 113 Maths Textbook Solution.

Answers (1)

Answer:a\left[\frac{x}{a} \tan ^{-1} \sqrt{\frac{x}{a}}+\tan ^{-1} \sqrt{\frac{x}{a}}-\sqrt{\frac{x}{a}}\right]+C

Hint: to solve this statement we will assume sin as tan

Given:   \int \sin ^{-1} \sqrt{\frac{x}{a+x}} d x

Solution:

\mathrm{I}=\int \sin ^{-1} \sqrt{\frac{x}{a+x}} d x

\text { Put } x=a \tan ^{2} t=>\tan t=\sqrt{\frac{x}{a}}

d x=a\left(2 \tan (t) \sec ^{2} t d t\right.

I=\int \sin ^{-1} \sqrt{\frac{\operatorname{atan}^{2} t}{a+\operatorname{atan}^{2} t}} a\left(2 \tan (t) \sec ^{2} t d t\right.

I=2 a \int \operatorname{ttan}(t) \sec ^{2} t d t

\text { Let t be the first function and } \tan (t) \sec ^{2} t d t \text { be the} e \text { second, applying Byparts }

I=a\left(\operatorname{ttan}^{2} t-\tan t+t\right)+C

I=a\left(\tan ^{2} t+t-\tan t\right)+C

I=a\left[\frac{x}{a} \tan ^{-1} \sqrt{\frac{x}{a}}+\tan ^{-1} \sqrt{\frac{x}{a}}-\sqrt{\frac{x}{a}}\right]+C

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