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JH
Numerade Educator

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Problem 13 Easy Difficulty

Evaluate the integral.

$ \displaystyle \int \sqrt{\cos \theta} \sin^3 \theta d \theta $

Answer

$\frac{-2}{5}(\cos \theta)^{5 / 2}+\frac{2}{7}(\cos \theta)^{7 / 2}+C$

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Video Transcript

This problem is from Chapter seven section to problem number thirteen in the book Calculus Early. Transcendental. Lt's a condition by James Store. Here we have the integral of the squared of co sign of data times San Cute of data. So first, let's rewrite this co sign this crude. Of course. Sinus cosign data to the one had power. Then for the sign. Cute. Let's re write this sine squared data time signed, Ada. Then we could use a pathetic an identity to rewrite sine squared as one minus co sign squared. So we have an integral coasts and data to the one half Power one minus co sign squared data times signed data and now we see we're ready to apply u substitution. So here we should take you'd be co sign of data. So that to you is negative scientist the equivalently negative. Do you? It's scientific data. So after applying to see substitution, we can rewrite this integral as thie integral of you to the one half power. Yeah, one minus you square, do you and we need a negative to you. So let's pull this negative outside of the rule. So now let's distribute issue to the one half power inside the apprentices. Whether you two, the three half power my tissue to the five over to power. And now we could evaluate those inner girls. And we have you threw the five paths times two over five, minus you seven halves types to over seven. And let's at the constant C of integration. And then here the last step is to replace you with co sign a potato. And we could also distribute the negative signs as well. So we have a negative two over five cosign data to the five half power, plus to over seven cosign data to the seven half power plus see, and that's your final answer.