Calculus II - Integrals Involving Trig Functions

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Section 7.2 : Integrals Involving Trig Functions

8. Evaluate \( \displaystyle \int{{\cos \left( {3t} \right)\sin \left( {8t} \right)\,dt}}\).

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There really isn’t all that much to this problem. All we have to do is use the formula given in this section for reducing a product of a sine and a cosine into a sum. Doing this gives,

\[\int{{\cos \left( {3t} \right)\sin \left( {8t} \right)\,dt}} = \int{{\frac{1}{2}\left[ {\sin \left( {8t - 3t} \right) + \sin \left( {8t + 3t} \right)} \right]\,dt}} = \frac{1}{2}\int{{\sin \left( {5t} \right) + \sin \left( {11t} \right)\,dt}}\]

Make sure that you pay attention to the formula! The formula given in this section listed the sine first instead of the cosine. Make sure that you used the formula correctly!

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Now all we need to do is evaluate the integral.

\[\int{{\cos \left( {3t} \right)\sin \left( {8t} \right)\,dt}} = \frac{1}{2}\left( { - \frac{1}{5}\cos \left( {5t} \right) - \frac{1}{{11}}\cos \left( {11t} \right)} \right) + c = \require{bbox} \bbox[2pt,border:1px solid black]{{ - \frac{1}{{10}}\cos \left( {5t} \right) - \frac{1}{{22}}\cos \left( {11t} \right) + c}}\]