First slide

Methods of integration

Question

The function $f$ whose graph passes through $(π / 4, 0)$
and whose derivative is $\frac{\log (\tan x)}{\sin x \cos x}$ is given by

Easy

A

$(\log \tan x)^{2} + \frac{π}{4}$
B

$\frac{1}{2} (\log \cos x)^{2}$
C

$\frac{1}{2} (\log \sin x)^{2}$
D

$\frac{1}{2} (\log \tan x)^{2}$

Solution

$\begin{array}{l} f (x) = \int \frac{\log (\tan x)}{\sin x \cos x} = \frac{1}{2} (\log \tan x)^{2} + C \\ 0 = f (\frac{π}{4}) = \frac{1}{2} (\log 1)^{2} + C \Rightarrow C = 0 \end{array}$

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