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Evaluate the integral.

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

$\frac{\sin \left(t^{2}\right)}{2}-\frac{\sin ^{3}\left(t^{2}\right)}{3}+\frac{\sin ^{5}\left(t^{2}\right)}{10}+C$

Integration Techniques

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This problem's from Chapter seven section to Problem number six in the book Calculus Early Transcendental Sze In addition by James Stored for this problem, we have an indefinite noble of tee times Co signed the fifth power of T Square. So here to suggest that we should do a u substitution to start us off. Let's take you evils. He's square so that do you is to t d t by the power rule. Yeah, since there's no too in front of the T and the integral, we should multiply this by one half. To get one half to you equals Titi Okay, so if we do this, we can rewrite this in Abril is one half co signed to the fifth power of you. Do you now? Since the power on the co sign is odd, let's re write this by pulling on one factor of co sign. So we have one half in a rule coastline to the fourth power of you times cose on you to you. And now let's deal with this coastline to the four power term. So if I come over here, let's go to the side and rewrite this so we can rewrite coastline to the fourth simply as co sign Squared Square. And now let's apply a dragon identity to reread it. Co sign squared is one minus. Science for review Square the whole thing. And then we can evaluate this latest expression by distribution to get one minus to science for interview plus sign. So the fourth hour of you so that will allow us to re write this in a rule is one half in a girl. So we'll replace this close into the fourth Power with our latest expression over here. Use from apprentices for this one minus to sign Square of You, plus signs to the fourth power view. And let's not forget this co signed you do at the end. All right, let's separate this from Earth scratch work on the side, and we could see our latest Integral. It's set up for another U substitution. So here, let's do another substitution on the side. Let's not use the letter W. Since that was already used above for original substitution to U equals C Square. So this time let's just introduce another very well. It's a w w equals sign of you so that t W becomes coz I knew Do you? So if we apply this new U substitution on this w sub, we had one half in general one minus two w square plus W to the fourth power D w. And this could be evaluated by using the power rule for each of these three terms. So we have one half w minus two w cute over three plus W to the fifth hour over five. Let's add our constancy of integration here at the end. So now we evaluated in a girl. But we should rewrite this so that it's an original variable C not w So one step would be to go from w back into you using your substitution so we can rewrite. This is sign you minus two sang cubed you over three plus signed to the fifth hour of you over five plus e and we're still not back in our variable t. So once more will come back here and backs up from you to T Square so they get a one half science. He's where minus two Sign cubed T squared over three plus signs of the fifth power We have a He's square all over five plus c and that's your final answer