# ∫ E X [ Sec X + Log ( Sec X + Tan X ) ] D X - Mathematics

Sum
$\int e^x \left[ \sec x + \log \left( \sec x + \tan x \right) \right] dx$

#### Solution

$\text{ Let I } = \int e^x \left[ \sec x + \text{ log }\left( \sec x + \tan x \right) \right]dx$

$\text{ Here, } f(x) = \text{ log }\left( \sec x + \tan x \right) Put e^x f(x) = t$

$\Rightarrow f'(x) = \sec x$

$\text{ let e}^x \text{ log }\left( \sec x + \tan x \right) = t$

$\text{ Diff both sides w . r . t x }$

$e^x \text{ log }\left( \sec x + \tan x \right) + e^x \frac{1}{\sec x + \tan x}\left( \sec x + \tan x + \sec^2 x \right) = \frac{dt}{dx}$

$\Rightarrow \left[ e^x \log\left( \sec x + \tan x \right) + e^x \left( \sec x \right) \right]dx = dt$

$\Rightarrow e^x \left[ \sec x + \log\left( \sec x + \tan x \right) \right]dx = dt$

$\therefore \int e^x \left[ \sec x + \text{ log} \left( \text{ sec x} + \tan x \right) \right]dx = \int dt$

$= t + C$

$= e^x \text{ log }|\left( \sec x + \tan x \right) + C$

Concept: Indefinite Integral Problems
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#### APPEARS IN

RD Sharma Class 12 Maths
Chapter 19 Indefinite Integrals
Exercise 19.26 | Q 8 | Page 143