In this video I go over a summary of the guidelines to help solve integrals of the form sin(x)^m cos(x)ⁿ where n and m are integers greater than or equal to 0. This video summarizes the example that I covered in my earlier videos and gives the appropriate steps in evaluating integrals of this type.

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Trigonometric Integrals: Guidelines for sin(x)cos(x)

To summarize on the examples I covered earlier, the following guidelines help in evaluating integrals of the form:

Where m ≥ 0 and n ≥ 0 are integers.

Strategy

(a) If the power of cosine is odd (n = 2k + 1), save one cosine factor and use cos²x = 1 - sin²x to express the remaining factors in terms of sine:

Then substitute u = sin x because du = cos x dx

(b) If the power of sine is odd (m = 2k + 1), save one sine factor and use sin²x = 1 - cos²x to express the remaining factors in terms of cosine:

Then substitute u = cos x because du = - sin x dx.

(c) If the powers of both sine and cosine are odd then either steps (a) or (b) can be used.

(d) If the powers of both sine and cosine are even, use the half-angle identities:

It is sometimes helpful to use the identity: