The value of _ ^ e^x ^2 ( e^x ) ;dx is - Vidyadip

MCQ

The value of$\int_{}^{} {{e^x}{{\sec }^2}({e^x})\;dx} $ is

✓
$\tan ({e^x}) + k$
B
$\tan ({e^x})\;.\;e + k$
C
${e^x}\tan x + k$
D
$\frac{{\tan ({e^x})}}{{{e^x}}} + k$

✓

Answer

Correct option: A.

$\tan ({e^x}) + k$

a
(a) $I = \int_{}^{} {{e^x}{{\sec }^2}({e^x})\,dx} $

Put ${e^x} = t \Rightarrow {e^x}dx = dt$
$\therefore \,\,\,I = \int_{}^{} {{{\sec }^2}t\,dt = \tan t + k = \tan ({e^x}) + k} $.

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