# How do you integrate #int (25+x^2)/sqrt(x^2-4)dx# using trigonometric substitution?

Use now the trigonometric identity:

and:

using the linearity of the integral:

This are fairly known integrals, but we can go through the solution:

and:

Putting it together:

and undoing the substitution, considering that:

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To integrate (\int \frac{{25 + x^2}}{{\sqrt{{x^2 - 4}}}} dx) using trigonometric substitution, follow these steps:

- Recognize the form of the integral and the need for trigonometric substitution.
- Substitute (x = 2\sec(\theta)), which implies (dx = 2\sec(\theta) \tan(\theta) d\theta), and simplify the expression.
- Express (x^2) in terms of (\theta) using the substitution.
- Rewrite the integral using trigonometric identities involving secant and tangent.
- Simplify the expression by factoring and canceling common terms.
- Integrate the resulting expression in terms of (\theta).
- Express the result in terms of (x) by substituting back the expression for (\theta).
- Simplify the final expression if necessary.

Following these steps, the integral can be evaluated using trigonometric substitution.

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To integrate (\int \frac{25 + x^2}{\sqrt{x^2 - 4}} , dx) using trigonometric substitution, let (x = 2\sec(\theta)). Then, (dx = 2\sec(\theta)\tan(\theta) , d\theta). After substitution and simplification, the integral becomes (\int \frac{25 + 4\sec^2(\theta)}{2\tan(\theta)} , (2\sec(\theta)\tan(\theta)) , d\theta). Simplify further to obtain (\int (25\sin(\theta) + 2\sec(\theta)\tan^2(\theta)) , d\theta). Finally, integrate term by term to get the result.

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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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