How do you integrate #int 1/(x^3sqrt(9-x^2))dx# using trigonometric substitution?
Whenever you have an integral with a
So, let
#=>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
#=>-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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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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To integrate ( \frac{1}{x^3\sqrt{9-x^2}} ) using trigonometric substitution, follow these steps:
- Substitute ( x = 3\sin(\theta) ).
- Calculate ( dx ) using the derivative of ( \sin(\theta) ).
- Rewrite the integral in terms of ( \theta ).
- Use trigonometric identities to simplify the expression.
- Integrate the simplified expression with respect to ( \theta ).
- 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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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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