# How do you integrate #int 1/(x^4sqrt(x^2-7))# by trigonometric substitution?

See calculation below:

Continuation:

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To integrate ( \frac{1}{x^4 \sqrt{x^2 - 7}} ) by trigonometric substitution, follow these steps:

- Let ( x = \sqrt{7} \sec(\theta) ).
- Substitute ( x = \sqrt{7} \sec(\theta) ) into the integral.
- Express ( \sec(\theta) ) in terms of ( x ) and rewrite the integral in terms of ( \theta ).
- Compute ( dx ) in terms of ( \theta ) using the derivative of ( \sec(\theta) ).
- Substitute ( dx ) in terms of ( \theta ) into the integral.
- Simplify the integrand using trigonometric identities.
- Integrate the simplified expression with respect to ( \theta ).
- Once integrated, convert back to the variable ( x ).

This process should yield the solution to the integral.

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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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