## Integration of sine and cosine powers

In the section **reduction formulas**, we got the formulas:

In the case where n = 2, these formulas are:

Alternative forms for these integration formulas can be obtained using trigonometric identities.

that come from the formulas for the double angle

These identities give rise to

## Integration of sines and cosines products

If **m** and **no** are positive integers, so the integral

can be calculated in a number of ways depending on **m** and **no** be even or odd.

**Example**

Calculate

**Solution.**

**Integration of tangent and secant powers**

The procedure for integrating tangent and secant powers follows in parallel those of sine and cosine. The idea is to use the following reduction formulas to reduce the integrand exponent until the resulting integral can be calculated:

In the case where *no* is odd, the exponent can be reduced to one, leaving us with the problem of integrating tg *x *or sec *x*.These integrals are given by

The formula can be obtained by writing

The formula requires a trick. We write

The following integrals occur frequently, and are worth noting:

Formula (2) has already been seen since the derivative of tgx is Formula (1) may be obtained by applying** The **reduction formula with *no*= 2, or alternatively, using the identity

to write

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