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Evaluate the integral.

$ \displaystyle \int t \sin^2 t dt $

$$\frac{1}{4} t^{2}-\frac{1}{4} t \sin 2 t-\frac{1}{8} \cos 2 t+C$$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 2

Trigonometric Integrals

Integration Techniques

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this problem is from Chapter seven section to problem number nineteen of the book Calculus Early transcendence, ALS eighth edition by James Toe And here we have an indefinite girl of tea time Sign square of T. So the first thing we can do is applying double angle formula from trigonometry to rewrite sine squared as one minus coast line of duty. All over too. We could pull out this one half outside of the integral and we could also this should be to t through the apprentices. This first intern girl, we can evaluate using the Potter rule for the second Integral will need integration. My parts. So let's split this into two on the rules. So everyone half and a girl of tea using the power rule is just t squared over two and then we have a minus one half in general t coastline of two tea dt So we'LL need in English my part here we can take you too, Bt so that d was dt and we can also take Do you need to be cose an opportunity? What titi So that be is sign off duty over two Our inner world becomes T squared over four minus one half and then applying the integration by parts formula. The first term is you times he So we have a tee times sign of two tea over two, minus the integral of the times. Do you? So that's a sign of two tea over to Titi. So let's simplify this t score over four minus t sign of two tea over four. And then here we have these double negative that cancels out and we have these twos in the denominator that will give us a one force and then we have an integral sign of two tea. DT. So we have one more integral to compute and for this integral and my help to go ahead and use the use of your vehicles to tea. So, doing this, we have he squared over four t sign to tear before plus one over four negative call sign of two t over too, plus e. And we could simplify this last expression by just pulling out that minus and multiplying the two in the four. So we have ah one minus one over eight cosign to tea. Plus he and that's our answer

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