First slide

Introduction to integration

Question

$\int \sin^{3} {xcos}^{3} xdx$ is equal to

Easy

A

$\frac{1}{32} [- \frac{3}{2} \cos 2 x + \frac{1}{6} \cos 6 x] + C$
B

$\frac{1}{16} [- \frac{3}{2} \cos 2 x + \frac{1}{6} \cos 6 x] + C$
C

$\frac{1}{64} [- \frac{3}{2} \cos 2 x + \frac{1}{6} \cos 6 x] + C$
D

None of the above

Solution

$\begin{array}{l} \int \sin^{3} {xcos}^{3} xdx = \frac{1}{8} \int (2 \sin xcos x)^{3} dx \\ = \frac{1}{8} \int \sin^{3} 2 xdx = \frac{1}{8} \int \frac{3 \sin 2 x - \sin 6 x}{4} dx \\ = \frac{1}{32} \int (3 \sin 2 x - \sin 6 x) dx \\ = \frac{1}{32} [- \frac{3}{2} \cos 2 x + \frac{1}{6} \cos 6 x] + C \end{array}$

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