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Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 50

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Answer: e^{\tan^{-1}x}+c

Hints: You must know about the integral rule of exponential and trigonometric functions.

Given: \int \frac{e^{\tan^{-1}x}}{1+x^{2}}dx

Let\tan^{-1}x =t and differentitate on both sides,

\begin{aligned} &\frac{1}{1+x^{2}}=\frac{d t}{d x} \\ &d x=\left(1+x^{2}\right) d t \\ &\int \frac{e^{t}}{1+x^{2}} \cdot\left(1+x^{2}\right) d t \\ &\int e^{t} d t=e^{t}+c \\ &=e^{\tan ^{-1} x}+c \quad\quad\quad\quad\quad\left[\int e^{x} d x=e^{x}\right] \end{aligned}

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