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Problem 45 Hard Difficulty

Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take $ C = 0 $).

$ \displaystyle \int x^3 \sqrt{1 + x^2} dx $


$\frac{1}{5}\left(1+x^{2}\right)^{5 / 2}-\frac{1}{3}\left(1+x^{2}\right)^{3 / 2}+C$


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Video Transcript

The problem is you wanna wait? It's a definite integral illustrate. And the check at your Gunther is reasonable grabbing boasts of function. Pan is untied. Preventive off The definite integral is integral acts hell, the tabs who tiff one plus x squared. But this problem first we can use new substitution. But you know they caught you Juan Class X squared then do you? Is the cultural to X now? Just indefinite integral. A stick or two into Girl X squared is because you want one and tax The axe is they got to weigh half you. We have a house for my half. This is the root of you. You now we can write. This's equal to integral of have you two three over? She was power into you minus I'm into girl. You too, huh? You This's equal to half terms. Two over you. Two off over. Two minus one half times, two over three halves. Huge. You three over to class casting the number. See, this is a natural one. Over. You two. I'm over two months. One through three. You two three over two plus. See then babe plugging one plus x squared to you how this's over. Five. Love us X Square, Uh, who are to one salon owner three and one Prospect Square three over two plus. See this is on high derivative of the function. Ax to hell has a beautiful one plus X squared Now by using some graphic. Otherwise we can just graph off this two functions as follows. There's a bloke here is entitled motive of the right care. We're now the determinative Off broke half is the right curve. So So, while you have to write curve at some point is the exactly slow puts a tan line off the blue curve at this point. Now, first, a signal captain. Too slow Port attended line of the blue curve. One X goes from negative infinity to infinity. We can see the slope of the tangent line. Half the brokers goes from negative infinity. Teo, Teo, Can you two have to be in plenty? Now let's look at why do you have the right to care of one axis goes from there to give the infinity to infinity. So why are you after right curve? Because from next to the affinity to zero and to infinity, since they are the same. So our result is very reasonable