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JH
Numerade Educator

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Problem 17 Easy Difficulty

Evaluate the integral.

$ \displaystyle \int \sin^2 x \sin 2x dx $

Answer

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

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Video Transcript

this problem is from Chapter seven section to Problem number 17 in the book Calculus Early Transcendental eighth Edition by James Stewart And here we have a indefinite integral of sine squared of X times sine of two x. So first thing we can do is apply a double angle formula to rewrite sign of two X So we have integral sign square eggs and then using our our double angle formula for sign. We have two sin x co sign X, the X so we can pull this to outside the integral and combine these signs to get assigned Cube. And this would be a good time to apply a u substitution. Here we can take you to be cynics so that do you is co sign of X dx. So if we use this use sub are integral becomes two times integral of u cubed, do you? We can evaluate this using the power rule for the integral two times U to the fourth power over four plus c. So here we can back substitute from you back to Synnex using our our u substitution. And this to over four becomes a one half. So we end up with signed to the fourth power of X over to plus C. And there's our answer