First slide

Methods of integration

Question

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

Moderate

A

$T a n (x e^{x}) + C$
B

$C o t (x e^{x}) + C$
C

$C o s (x . e^{x}) + C$
D

$S i n (x . e^{x}) + C$

Solution

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

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