First slide

Methods of integration

Question

$\int \cot (e^{x}) \cdot \cos e c^{2} (e^{x}) \cdot e^{x} d x is equal to$

Moderate

A

$- 2 \cot (e^{x}) + C$
B

$2 \cot (e^{x}) + C$
C

$\frac{- 1}{2} {(\cot (e^{x}))}^{2} + C$
D

$\frac{1}{2} \cot (e^{x}) + C$

Solution

$Let I = \int \cot (e^{x}) \cdot \cos e c^{2} (e^{x}) \cdot e^{x} d x$
$Let e^{x} = y$
$\begin{array}{l} e^{x} d x = d y \\ I = \int \cot y \cdot \cos e c^{2} y d y \end{array}$
$Put \cot y = z$
$\begin{array}{l} \cos e c^{2} y d y = - d z \\ I = \int z \cdot (- d z) \\ = \frac{- z^{2}}{2} + C \\ = - \frac{1}{2} (\cot y)^{2} + C \\ = - \frac{1}{2} {(\cot e^{x})}^{2} + C \end{array}$

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