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Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 57

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Answer:f(x)=\sec x

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


\int e^{x}\left ( \tan x+1 \right )\sec xdx=e^{x}f\left ( x \right )+c, find f(x)


I=\int e^{x}\left ( \tan x+1 \right )\sec xdx


\begin{aligned} &I=\int e^{x}(\tan x+1) \sec x d x \\ &=\int e^{x}(\sec x \tan x+\sec x) d x \\ &=\int e^{x} \sec x d x+\int e^{x} \sec x \tan x d x \end{aligned}

In second integral, apply integration by parts

     \begin{aligned} &\quad=\int e^{x} \sec x d x+e^{x} \int \sec x \tan x d x-\int \frac{d}{d x} e^{x}\left[\int(\sec x \tan x) d x\right] d x \\ &=e^{x} \sec x-\int e^{x} \sec x d x+\int e^{x} \sec x d x \\ &\quad \begin{aligned} &=e^{x} \sec x+c \\ &e^{x} \sec x+c=e^{x} f(x)+c \end{aligned} \\ &\therefore f(x)=\operatorname{secx} \end{aligned}

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