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Integrate cos and sin squared using double angles

January 27th, 2010 admin Leave a comment Go to comments

The integration of cos2x and sin2x comes up quite a lot and an easy trick for finding them is to use cos(2x). You do this using the following identities:

sin^{2}(x) = \frac{1-cos(2x)}{2} and cos^{2}(x) = \frac{1+cos(2x)}{2}

These are derived from the formula for double cos as follows
cos(2x) = cos^{2}(x) - sin^{2}(x)
\Leftrightarrow cos(2x) = cos^{2}(x) - 1 + cos^{2}(x) by subsitituion of sin^{2}(x) = 1 - cos^{2}(x)
\Leftrightarrow cos^{2}(x) = \frac{1 + cos(2x)}{2}
and similarly for sin2x but using the substitution for cos2x instead of sin2x.

To use these identities we simply substitute them into the integral and find the integral as normal since we know
\int \! cos(2x) \, dx = \frac{1}{2}sin(2x) + c

so we get
\int \! cos^{2}(x) \,dx = \int \! \frac{1+cos(2x)}{2} \, dx = \frac{x}{2} + \frac{1}{4}sin(2x) + c
and
\int \! cos^{2}(x) \,dx = \int \! \frac{1-cos(2x)}{2} \, dx = \frac{x}{2} - \frac{1}{4}sin(2x) + c

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