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  • provide solution for RD Sharma maths class 12 chapter Indefinite Integrals exercise 18.26 question 18

provide solution for RD Sharma maths class 12 chapter Indefinite Integrals exercise 18.26 question 18

Answers (1)

Answer:
The correct answer is  e^{x} \sin ^{-1} x+c
Hint:

Using ILATE rule

\int e^{x}\left\{f(x)+f^{\prime}(x)\right\} d x=e^{x} f(x)+c

Given:

\int e^{x} \frac{\sqrt{1-x^{2}} \sin ^{-1} x+1}{\sqrt{1-x^{2}}} d x

Solution:

        I=\int e^{x} \frac{\sqrt{1-x^{2}} \sin ^{-1} x+1}{\sqrt{1-x^{2}}} d x

It is in the form of,

        \int e^{x}\left\{f(x)+f^{\prime}(x)\right\} d x

Where    f(x)=\sin ^{-1} x

        \int e^{x}\left\{f(x)+f^{\prime}(x)\right\} d x=e^{x} f(x)+c

So,

        \int e^{x}\left(\frac{1}{\sqrt{1-x^{2}}}+\sin ^{-1} x\right)=e^{x} \sin ^{-1} x+c

So the correct answer is  e^{x} \sin ^{-1} x+c

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