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

Answer 1

#-3/2sqrt(1-x^2/2)+1/6sqrt(1-(3/sqrt(2)x-sqrt2x^2)^2)+C#

Try the substitution: #x = sqrt2sin(u)#
This would mean #dx = sqrt2cos(u)dx#

Now put this into the integral and we get:

#int(2^(3/2)sin^3(u))/sqrt(4-4sin^2(u))sqrt2cos(u) du#
Tidying this up a little and using the trig -identity: #sin^2(x)+cos^2(x) =1#, we get:
#int(4sin^3(u))/(2sqrt(1-sin^2(x)))cos(u)du#
#=2intsin^3u/(sqrt(cos^2(u)))cos(u)du=4intsin^3(u)du#
At this point we need another trig identity . The trig identity that we will use is: #sin^3x = 3/4sinx - 1/4sin(3x)#.

Putting that into the integral we get:

#2int3/4sinu-1/4sin(3u)dx#

Which can now be integrated to obtain:

#-3/2cos(u) + 1/6cos(3u)+C#

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:

#-3/2sqrt(1-sin^2(u))+1/6sqrt(1-sin^2(3u))+C#

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

#sin(3u) = 3sin(u) - 4sin ^3(u)#

We can now rewrite the above expression as:

#-3/2sqrt(1-sin^2(u))+1/6sqrt(1-(3sin(u)-4sin^3(u))^2)+C#

And finally we can reverse the substitution to obtain:

#-3/2sqrt(1-x^2/2)+1/6sqrt(1-(3/sqrt(2)x-sqrt2x^2)^2)+C#
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Answer 2

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

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

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

  1. Let ( x = \sqrt{2} \sin(\theta) ).
  2. ( dx = \sqrt{2} \cos(\theta) , d\theta ).
  3. 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 ).
  4. 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) ).
  5. 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 ).
  6. Integrate with respect to ( \theta ).
  7. After integration, substitute back ( x = \sqrt{2} \sin(\theta) ).
  8. Simplify the result.
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Answer 3

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

  1. Let (x = \sqrt{2} \sin(\theta)).
  2. Find (dx) by taking the derivative of (x) with respect to (\theta) and substituting it in the integral.
  3. Substitute (x) and (dx) in terms of (\theta) in the integral.
  4. Simplify the expression using trigonometric identities.
  5. Integrate the simplified expression with respect to (\theta).
  6. 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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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.

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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