First slide

Introduction to integration

Question

If $f (x) = \cos x$ then $\int (2 (f (x))^{2} - 1) (4 (f (x))^{3} - 3$
$f (x)) d x$ is equal to

Easy

A

$\frac{1}{10} \sin 5 x + \frac{1}{2} \sin x + C$
B

$\frac{1}{5} \sin 5 x + \sin x + C$
C

$\frac{1}{15} \sin 5 x - \frac{1}{2} \sin x + C$
D

none of these

Solution

$\begin{array}{l} I = \int (2 \cos^{2} x - 1) (4 \cos^{3} x - 3 \cos x) d x \\ = \int \cos 2 x \cos 3 x d x \\ = \frac{1}{2} \int [\cos 5 x + \cos x] d x \\ = \frac{1}{10} \sin 5 x + \frac{1}{2} \sin x + C \end{array}$

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