First slide

Methods of integration

Question

$\int \frac{e^{x} (1 + x)}{\cos^{2} (e^{x} x)} dx$ is equal to

Moderate

A

$- \cot (e^{x} x) + C$
B

$\tan ({xe}^{x}) + C$
C

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

$\cot (e^{x}) + C$

Solution

$\begin{array}{l} \int \frac{e^{x} (1 + x)}{\cos^{2} (e^{x} x)} dx \\ Let {xe}^{x} = t \Rightarrow ({xe}^{x} + e^{x}) = \frac{dt}{dx} \Rightarrow dx = \frac{dt}{e^{x} (x + 1)} \\ ∴ \int \frac{e^{x} (1 + x)}{\cos^{2} (e^{x} x)} dx = \int \frac{e^{x} (1 + x)}{\cos^{2} t} \times \frac{dt}{e^{x} (1 + x)} \\ = \int \frac{1}{\cos^{2} t} dt = \int \sec^{2} tdt \\ = \tan t + C = \tan ({xe}^{x}) + C \end{array}$

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