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

The answer is

Perform this integral by substitution

The integral is

Therefore,

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To integrate ∫(1/(x^2*sqrt(4+x^2))) using trigonometric substitution, follow these steps:

- Recognize that the expression under the square root resembles a trigonometric identity, specifically the Pythagorean identity.
- Substitute x = 2*tan(θ) to simplify the integral.
- Express dx in terms of dθ using the derivative of the substitution.
- Rewrite the integral in terms of θ and evaluate it using trigonometric identities.
- Finally, substitute back the expression for θ in terms of x to obtain the final result.

If you require further clarification or assistance with any step, please feel free to ask.

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To integrate ( \int \frac{1}{x^2 \sqrt{4 + x^2}} ) by trigonometric substitution:

- Substitute ( x = 2 \tan(\theta) ).
- Express ( \sqrt{4 + x^2} ) in terms of ( \theta ).
- Substitute ( dx = 2 \sec^2(\theta) , d\theta ).
- Rewrite the integral in terms of ( \theta ).
- Simplify and integrate the expression with respect to ( \theta ).
- Finally, substitute back ( \theta ) in terms of ( x ) to get the final result.

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