First slide

Introduction to integration

Question

If $f (x) = \cos x$ and $g (x) = \sin x$ then $\int \frac{\log f (x)}{(f (x))^{2}} d x$ is

Easy

A

$f (x) (\log f (x) + 1) + x + C$
B

$\frac{g (x)}{f (x)} (\log f (x) + 1) + \frac{x^{2}}{2} + C$
C

$\frac{f (x)}{g (x)} (\log g (x) + 1) + x + C$
D

$\frac{g (x)}{f (x)} [\log f (x) + 1] - x + C$

Solution

$\begin{array}{l} I = \int \frac{\log \cos x}{\cos^{2} x} d x = \int \sec^{2} x \log \cos x d x \\ = (\tan x) \log \cos x - \int \tan x (\frac{- \sin x}{\cos x}) d x \\ = (\tan x) \log \cos x + \int (\sec^{2} - 1) d x \\ = (\tan x) [\log \cos x + 1] - x + C \end{array}$
$= \frac{g (x)}{f (x)} [\log f (x) + 1] - x + C$

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