Trigonometric functions appear in the natural world frequently, so understanding how to integrate them is essential to solving real problems. There are a number of situations we can have involving trigonometric function, and corresponding rules to solve them.

**Example: **Evaluate

If we tried using the substitution rule, we would have and , which doesnâ€™t really help. However, we can manipulate this function to be easier to work with. If we factor our a term, we would have

From here, we can use the trig identity, to change the into . This gives us the following integral.

From here, we can now apply the substitution rule to get a solution. If we let u = sin(x), then du = cos(x)dx, giving us the integral

When we have functions in the form , there is a set of rules we can follow to evaluate them correctly.

**Strategy for evaluating **

- If the power of cosine, n, is odd (n = 2k+1), save one factor of cosine and use to express the remaining factors in terms of sine: , then substitute u = sin(x)
- If the power of sine is odd (m = 2k+1), save one factor of sine and use to express the remaining factors in terms of cosine: , then substitute u = cos(x)
- If both the power of sine is odd and the power of cosine is odd, then either 1 or 2 can be used.
- If the powers of both sine and cosine are even, use the half angle identities or

We can also evaluate functions that include tan(x) and sec(x)

**Strategy for evaluating **

- If the power of secant is even (n = 2k), save a factor of , and use to express the remaining factors in terms of tan(x): , the substitute u = tan(x)
- If the power of tangent is odd (m = 2k+1), save a factor of sec(x)tan(x) and use to express the remaining factors in terms of sec(x):

, then substitute u = sec(x).

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