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

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

Answers (1)

Answer:f(x)=\sec x

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

Given:

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

Solution:

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

Consider,

\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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