# How do you integrate #int dx/(5-4x-x^2)^(5/2)# by trigonometric substitution?

Complete the square at the denominator:

Subtitute:

Then:

and as:

we have:

To undo the substitution note that:

so:

Then:

and simplifying:

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To integrate ( \int \frac{{\text{d}x}}{{(5 - 4x - x^2)^{5/2}}} ) using trigonometric substitution:

- Complete the square in the denominator to rewrite the expression in terms of a perfect square.
- Apply trigonometric substitution by letting ( x = \sqrt{5}\sin(\theta) ) or ( x = \sqrt{5}\cos(\theta) ), depending on the quadratic expression under the square root.
- Express ( \text{d}x ) in terms of ( \theta ).
- Substitute ( x ) and ( \text{d}x ) in terms of ( \theta ) into the original integral.
- Simplify the expression using trigonometric identities.
- Integrate the resulting expression with respect to ( \theta ).
- Finally, substitute back ( \theta ) in terms of ( x ) to obtain the final answer in terms of ( x ).

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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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