First slide

Methods of integration

Question

$\int \frac{\ln (\tan x)}{\sin x \cdot \cos x} d x, is equal to$

Easy

A

$\frac{1}{2} \ln (\tan x) + C$
B

$\frac{1}{2} \ln (\tan^{2} x) + C$
C

$\frac{1}{2} (\ln (\tan x))^{2} + C$
D

$None of these$

Solution

$\begin{array}{l} I = \int \frac{\ln (\tan x)}{\sin x \cdot \cos x} dx, let t = \ln (\tan x) \\ \Rightarrow \frac{dt}{dx} = \frac{\sec^{2} x}{\tan x} \Rightarrow dt = \frac{dx}{\sin x \cdot \cos x} \\ ∴ I = \int tdt = \frac{1}{2} t^{2} + C = \frac{1}{2} (\ln (\tan x))^{2} + C \end{array}$

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