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How do you integrate: #(x^2 * dx) / sqrt(9 - x^2)#?

Answer 1

Ezra Adams

Try with the classical substitution #x=3sin(theta)#:
So:

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Answer 2

Christopher Agresta

To integrate ( \frac{x^2}{\sqrt{9 - x^2}} ) with respect to ( x ), you can use trigonometric substitution.

Let ( x = 3\sin(\theta) ), then ( dx = 3\cos(\theta) , d\theta ).

Substitute ( x ) and ( dx ) into the integral, and ( \sqrt{9 - x^2} ) becomes ( 3\cos(\theta) ).

The integral becomes ( \int \frac{(3\sin(\theta))^2}{3\cos(\theta)} \cdot 3\cos(\theta) , d\theta ).

Simplify to ( \int 9\sin^2(\theta) , d\theta ).

Using the identity ( \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} ), the integral becomes ( \int \frac{9}{2} - \frac{9\cos(2\theta)}{2} , d\theta ).

Integrate term by term to get ( \frac{9}{2}\theta - \frac{9}{4}\sin(2\theta) + C ).

Finally, substitute back ( \theta = \sin^{-1}\left(\frac{x}{3}\right) ) to get the answer in terms of ( x ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.