# How do you integrate #int x^3 /sqrt(4 - 2x^2) dx# using trigonometric substitution?

Now put this into the integral and we get:

Putting that into the integral we get:

Which can now be integrated to obtain:

We obviously have the challenge now of reversing the substituting. Again using the trig identity near the start, this formula can be re - arranged to give:

At this point we can now rearrange the above trig identity to get:

We can now rewrite the above expression as:

And finally we can reverse the substitution to obtain:

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To integrate ( \int \frac{x^3}{\sqrt{4 - 2x^2}} , dx ) using trigonometric substitution:

- Let ( x = \sqrt{2} \sin(\theta) ).
- Then, ( dx = \sqrt{2} \cos(\theta) , d\theta ).
- Substitute ( x ) and ( dx ) in terms of ( \theta ) into the integral.
- Rewrite the expression under the square root using trigonometric identities.
- Perform the substitution and simplify.
- Integrate with respect to ( \theta ).
- Substitute back for ( x ) using the original substitution.
- Simplify the result.

The detailed steps for solving the integral using trigonometric substitution are as follows:

- Let ( x = \sqrt{2} \sin(\theta) ).
- ( dx = \sqrt{2} \cos(\theta) , d\theta ).
- Substitute into the integral: ( \int \frac{(\sqrt{2}\sin(\theta))^3}{\sqrt{4 - 2(\sqrt{2}\sin(\theta))^2}} \cdot \sqrt{2} \cos(\theta) , d\theta ).
- Simplify the expression under the square root: ( 4 - 2(\sqrt{2}\sin(\theta))^2 = 4 - 2 \cdot 2 \sin^2(\theta) = 4 - 4\sin^2(\theta) = 4\cos^2(\theta) ).
- Substitute into the integral and simplify: ( \int \frac{2\sqrt{2} \sin^3(\theta)}{\sqrt{4} \cdot |\cos(\theta)|} \cdot \sqrt{2} \cos(\theta) , d\theta ). ( = \int \frac{4\sqrt{2} \sin^3(\theta) \cos(\theta)}{|\cos(\theta)|} , d\theta ).
- Integrate with respect to ( \theta ).
- After integration, substitute back ( x = \sqrt{2} \sin(\theta) ).
- Simplify the result.

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To integrate (\int \frac{x^3}{\sqrt{4 - 2x^2}} dx) using trigonometric substitution:

- Let (x = \sqrt{2} \sin(\theta)).
- Find (dx) by taking the derivative of (x) with respect to (\theta) and substituting it in the integral.
- Substitute (x) and (dx) in terms of (\theta) in the integral.
- Simplify the expression using trigonometric identities.
- Integrate the simplified expression with respect to (\theta).
- Substitute back (x) in terms of (\theta) to obtain the final result.

After integrating, the result will be:

(\frac{\sqrt{2}}{8} \left(\theta + \frac{1}{2}\sin(2\theta)\right) + C), where (C) is the constant of integration.

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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.

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.

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.

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