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Evaluate the integral.

$ \displaystyle \int \sin^5 (2t) \cos^2 (2t) dt $

$$-\frac{1}{14} \cos ^{7}(2 t)+\frac{1}{5} \cos ^{5}(2 t)-\frac{1}{6} \cos ^{3}(2 t)+C$$

Integration Techniques

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This problem is from Chapter seven section to Problem number five in the book Capitalists Early Transcendental Sze eighth Edition by James Story Here we have ah, indefinite Integral science of the fifth power of two t co sign squared of two tea. So here, since the power of sign is odd, the first thing we do is rewrite this by pulling out a factor of signs to have signed in the fourth Coastline Square. And let's put that extra factor of sign over here at the end. Chris, observe here in the inner grand that the science of the fourth power. So it's going to decide and rewrite this so we can rewrite sign square science of the four power as Science Square Square. Then we could apply, uh, put that in identity to rewrite sine squared is one minus coastlines where, and we have to square this entire expression and finally weaken. Evaluate this latest expression two co sign squared to team plus coastlines in the fourth power two teams, so replacing science of the forth. With this newest expression, we have the integral one. Minus two co signed square plus coast under the fourth Power Time's Cose identity to co sign squared to tease and sign off duty. So here it looks like we can apply u substitution here we should take you two be co sign opportunity, which implies to you is negative, too. Sign two t d t. That's using the chain rule and here because we don't have a negative to in front of the sign and the original under grand, we should make up for this, but by multiplying by a negative one half in front of the to you to give us exactly what we want. Sign two tea Titi Just as appears an immigrant. So now let's rewrite this integral using our U substitution. So first, let's pull off this negative on half outside the integral level one minus to use Claire. Plus, you know, the fourth then co sign squared is just use Claire and then to you. So now let's just leave this negative one half outside for now. And what's the stripy tissue squared inside the apprentices. So we have a use clear minus to you to the fourth power. Plus, you know the city's power to you to evaluate the center girl. We should supply the power rule three times. So we have you to the third power over three minus to you to the fifth over five. Plus, you're the seven over seven, plus our constancy of integration. So here we have two steps left will replace you with co sign of two tea, and we could multiply by this negative one have so we get a negative cool sign. Cube's Tootie over six. Here we have a plus coastline to the fifth hour over five. Those two's cancel. Now we have a minus co sign to the seventh hour, all divided by fourteen and closer constancy, and that's our answer.