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JH
Numerade Educator

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Problem 54 Easy Difficulty

Evaluate the integral.

$ \displaystyle \int (x + \sin x)^2\ dx $

Answer

$\frac{x^{3}}{3}-2 x \cos x+2 \sin x+\frac{x}{2}-\frac{\sin 2 x}{4}+C$

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Video Transcript

Let's start off by just expanding this into Grand X Square to Excite X Science, where then we could go ahead and split this sense of three generals. Oops, think of the ex first and then in a gruel of science squares. So for the first one, just use the power room here. Let's use and enrich my parts. Let's take you to be x two u equals the x t me signed the eggs. So he is negative. Cosign x sound. Recall the formula to be replaced the integral with UV minus in rule be, do you? So we have plus and then two times. So now we do you times V So this times this that's negative. X co sign X and then minus in a rule z Do you? So I have a negative co sign. Let me just go ahead and cancel those minuses. And then finally, for the last integral. It's used a half angle formula for sign, so it's one minus co sign to X, and we pulled out the two. So let's simplify this integral of co sign a sign. Don't forget the two out here. So here, in a goal of one X. Don't forget the one half hour in the front, out and then the end. Roll over here that'LL be a minus sign, too. It's over, too. But then we have another two over here. So as a four plus our constant of integration, see? And that's your final answer.