First slide

Introduction to integration

Question

$\int \frac{\sin^{4} x}{\cos^{2} x} d x = f (x) + \frac{1}{4} g (x) - \frac{3 x}{2} + c then g (x) / f (x) = ?$

Moderate

A

$2 \cos^{2} x$
B

$2 \sin^{2} x$
C

$2 \tan x$
D

$2 \cot x$

Solution

$\begin{array}{l} \int \frac{\sin^{4} x}{\cos^{2} x} d x = \int \frac{{(1 - \cos^{2} x)}^{2}}{\cos^{2} x} d x \\ = \int \frac{1 - 2 \cos^{2} x + \cos^{4} x}{\cos^{2} x} d x \\ = \int \sec^{2} x - 2 + \cos^{2} x d x \\ = \int \sec^{2} x - 2 + \frac{1 + \cos 2 x}{2} d x \\ = \int \sec^{2} x - \frac{3}{2} + \frac{1}{2} \cos 2 x d x \\ = \tan x - \frac{3 x}{2} + \frac{1}{4} \sin 2 x + c \\ f (x) = \tan x, & g (x) = \sin 2 x \end{array}$

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