Name: Antonio
Who is asking: Student

Question:
Hi. Can you please tell me how to integrate a trig function involving sine and cosine? I know if the powers of both the sine and cosine are even and nonnegative, then I can make repeated use of the power-reducing formulas. But for the question I have on my hand, the powers of both sine and cosine are odd:
( sin³x + cos⁷x ) dx.
How can I tackle this question?
Thank you for answering

Hi Antonio,

If a sine in raised to an odd power, for example sin³x as in your example, the technique is to separate one factor of sin x and then write the remaining factor, an even power of sin x, in terms of cos x, that is

sin³x = sin²x sin x = (1-cos²x) sin x

This can be integrated with a substitution of u = cos x.

The second term can be integrated in a similar fashion

cos⁷x = cos⁶x cos x = (cos²x)³ cos x = (1-sin²x)³ cos x

Harley

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