If int e^sec x (sec xtan x f(x)+ (sec xtan x+sec^2x ) )dx=e^sec xf(x)+C , then a possible choice of f(x) is :

Indefinite Integration: If int e^sec x (sec xtan x f(x)+ (sec xtan x+sec^2x ) )dx=e^sec… | JEE Main 2019

JEE Main 2019 (Online) 9th April Evening ShiftQuestion 1

If $\int e^{\sec x}\left(\sec x\tan x\,f(x)+\left(\sec x\tan x+\sec^{2}x\right)\right)dx=e^{\sec x}f(x)+C$, then a possible choice of $f(x)$ is :

Indefinite Integration

$\sec x-\tan x-\frac12$$\sec x+\tan x+\frac12$✓ Correct answer$x\sec x+\tan x+\frac12$$\sec x+x\tan x-\frac12$

Solution

Concept. Differentiate the right side and compare with the integrand. Formulas used. (1) $\dfrac{d}{dx}\left(e^{\sec x}f(x)\right)=e^{\sec x}\left(\sec x\tan x\,f(x)+f'(x)\right)$Step 1 - Compare. By (1), the integrand matches when $$f'(x)=\sec x\tan x+\sec^2x.$$ Step 2 - Integrate. $f(x)=\sec x+\tan x+k$ for any constant $k$. Step 3 - Choose. With $k=\frac12$: $f(x)=\sec x+\tan x+\frac12$. Answer: (B)

Indefinite Integration - JEE Mathematics question (JEE Main 2019 (Online) 9th April Evening Shift)

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