Problem 43 Medium Difficulty

Evaluate the integral.

$ \displaystyle \int_0^{\frac{\pi}{2}} \cos 5t \cos 10t dt $

Answer

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Video Transcript

this problem is from Chapter seven section to problem number forty three in the book Calculus Early Transcendental Sze eighth Edition by James Door We have a definite in a rule of co signed five tee times Cose I intent e here it will be useful to use a formula for the product of two co science. In our case, we have a formula from the text and in our problem, the value of a five t and we could take b equals tensy. So using this formula pulling out the one half we have one half Indian girl zero power too coastline of a minus B So it's a five t minus twenty plus coastline of a plus B. So it's five. Teen plus twenty. Okay, so observing the first co sign here we have a minus five t so we can replace co sign of minus five t with co sign a five t since coastline is he even function not necessary to do the step But it will simplify things a little bit for us. So we have one half and a girl zero power too cosign five t plus coastline fifteen t and my help you to use the U substitution here to evaluate these inaugurals. You could take you to the five T and here you can take you two be fifteen t. Yeah. Evaluating these generals, we have one half signed by T over five. Plus I kn fifteen t over fifteen and points are zero in power for too. Let's go ahead and plug those in for tea. So we have one half sign of five pie over, too. On princesses, plus sign of fifteen power, too. That's also in Prentice's over fifteen. So this is from plugging in Pi over till for team. And when we plug in zero for tea, we have sign of zero over five, plus sign zero over fifteen. And we know from the unit circle tables of these terms will be zero no, also from the unit circle. We know that sign of five pilots who is one. So we have one half. We have won over five and then sign of fifteen pirates who was minus one. So we have a negative one over fifteen, and this can be simplified to give us two over thirty, which is one over fifteen. And that's our answer