First slide

Methods of integration

Question

$Evaluate \int \frac{e^{x} (1 + x)}{\cos^{2} (x e^{x})} d x$

Easy

A

$\tan (x e^{x}) + c$
B

$\tan (e^{x}) + c$
C

$\tan (x + e^{x}) + c$
D

$s e c (e^{x}) + c$

Solution

$Let z = x e^{x} . Then d z = (1 e^{x} + x e^{x}) d x = e^{x} (1 + x) d x .$
$\begin{array}{l} T h u s, \int \frac{e^{x} (1 + x)}{\cos^{2} (x e^{x})} d x = \int \frac{d z}{\cos^{2} z} \\ = \int \sec^{2} z d z \\ = \tan z + c = \tan (x e^{x}) + c \end{array}$

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