How do you integrate #int 1/(x^3sqrt(9-x^2))dx# using trigonometric substitution?

Answer 1

#-1/54(ln|(3 + sqrt(9 - x^2))/x| + (3sqrt(9 - x^2))/x^2) + C#

Whenever you have an integral with a #√# of the form #sqrt(a^2 - x^2)#, use the substitution #x = asintheta#.

So, let #x = 3sintheta#. Then #dx= 3costhetad theta#.

#=>int 1/((3sintheta)^3sqrt(9 - (3sintheta)^2)) * 3costhetad theta#

#=>int 1/(27sin^3theta sqrt(9 - 9sin^2theta)) * 3costheta d theta#

#=>int 1/(27sin^3thetasqrt(9(1 - sin^2theta))) * 3costheta d theta#

#=>int 1/(27sin^3thetasqrt(9cos^2theta)) * 3costheta d theta#

#=>int 1/(27sin^3theta3costheta) * 3costheta d theta#

#=>int 1/(27sin^3theta) d theta#

#=>1/27int csc^3theta d theta#

This can be integrated as #-(ln|csctheta + cottheta| + cotthetacsctheta)/2 + C#. This is derived using integration by parts. You can see the full proof here.

#=>-1/54(ln|csctheta + cottheta| + cotthetacsctheta) + C#

Now draw a triangle to determine expressions for cosecant and cotangent.

#=>-1/54(ln|(3 + sqrt(9 - x^2))/x| + (3sqrt(9 - x^2))/x^2) + C#

Hopefully this helps!

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Answer 2

To integrate (\int \frac{1}{x^3\sqrt{9-x^2}}dx) using trigonometric substitution, let (x = 3\sin\theta), (dx = 3\cos\theta d\theta). Then:

[ \begin{align*} \int \frac{1}{x^3\sqrt{9-x^2}}dx &= \int \frac{1}{(3\sin\theta)^3\sqrt{9-(3\sin\theta)^2}} \cdot 3\cos\theta d\theta\ &= \int \frac{1}{27\sin^3\theta \cdot \sqrt{9(1-\sin^2\theta)}} \cdot 3\cos\theta d\theta\ &= \frac{1}{9}\int \frac{\cos\theta}{\sin^3\theta}d\theta\ &= \frac{1}{9}\int \csc^2\theta d\theta\ &= -\frac{1}{9}\cot\theta + C \end{align*} ]

Recall that (x = 3\sin\theta), so (\theta = \sin^{-1}\left(\frac{x}{3}\right)).

Therefore, (\int \frac{1}{x^3\sqrt{9-x^2}}dx = -\frac{\cot(\sin^{-1}(x/3))}{9} + C).

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Answer 3

To integrate ( \frac{1}{x^3\sqrt{9-x^2}} ) using trigonometric substitution, follow these steps:

  1. Substitute ( x = 3\sin(\theta) ).
  2. Calculate ( dx ) using the derivative of ( \sin(\theta) ).
  3. Rewrite the integral in terms of ( \theta ).
  4. Use trigonometric identities to simplify the expression.
  5. Integrate the simplified expression with respect to ( \theta ).
  6. Substitute back the original variable ( x ) for ( \theta ) to obtain the final result.

By following these steps, you can integrate the given expression using trigonometric substitution.

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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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