First slide

Methods of integration

Question

$\int \cot^{3} x d x =$

Moderate

A

$\frac{{(\cot x)}^{2}}{2} + \log (\sin x) + C$
B

$- 3 \cot^{2} x \cos e c^{2} x + C$
C

$3 \cot^{2} x \cos e c^{2} x + C$
D

$- \frac{{(\cot x)}^{2}}{2} + \log (\cos e c x) + C$

Solution

$\begin{array}{l} \underset{}{I = \int \cot^{3} x d x = \int \cot^{2} x \cot x d x} \\ I = \int ({cosec}^{2} x - 1) \cot x d x \\ = \int \cot x {cosec}^{2} x d x - \int \cot x d x \\ Let I_{1} = \int \cot x {cosec}^{2} x d x \\ \cot x = t \to - {cosec}^{2} x d x = d t \\ I_{1} = \int t d t = \frac{- t^{2}}{2} \\ I = - \frac{(\cot x)^{2}}{2} - \log | \sin x | + C \end{array}$

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