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JH
Numerade Educator

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Problem 43 Medium Difficulty

Evaluate the integral.

$ \displaystyle \int \frac{\sqrt{x}}{1 + x^3}\ dx $

Answer

$$\frac{2}{3} \tan ^{-1}\left(x^{3 / 2}\right)+C$$

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Video Transcript

Let's try use up here. Let's take you to just be the numerator. Then do you use the power rule here for derivatives? We can also write. This is one over to you, the ex and then solving this equation. The circle equation for DX just multiplied the to you to the left side. So here, let's rewrite this integral. Now, after using the use of so this up here, that's just you, from our new substitution and then on the denominator, we have one plus and then here we need to be careful. So raise both sides to the sixth power Here you could hear the six and then you get X to the one half to the six and that becomes execute. So we want you to the six here and then for DX. We used this equation over here and we replaced the ex with to you. Do you? Let's go ahead and pull out That too. Simplify the numerator and then here we should try another u substitution. Let's take VTB You killed then Devi over three is useful here. Do you so plugging this end? You're just the one up top and then we have one plus V squared on the bottom because these were equals you Cube square, which is, you know, the six. So that's why we replace you two, the six with these flare and then this integral you may have memorized by now. If not, you can use the tricks up to get either right answer. Just take me to be one times tangelo Zeta. And after doing this you get two thirds and then the integral becomes arc tan ten in verse, the let's add that constancy of integration. And then now let's go ahead and replace V with the definition of the use of that's you cute. And since I am running out of room here, let me go on to the next page. So we have two thirds ten in verse, you cubed, plus he and then recall the definition of you was radical X or, if you like X to the one has and this means you cubed is Exeter three house. So if we like we can replace this, you third inside the ten with X to the three over too, plus C. And that's your final answer