# Simple Integral w/ Trigonometric Substitution

In summary, the conversation discusses solving the integral \int \sqrt{x^2 - 1} dx using the substitution x = sec \theta and integrating by parts. It is mentioned that the integral \int tan^2 \theta sec \theta d \theta can be simplified and solved using a substitution, while the integral \int sec^3 \theta d\theta requires multiple uses of integration by parts.
Hello everyone, I am having some trouble with an integral.
$$\int \sqrt{x^2 - 1} dx$$

so far:
$$x = sec \theta$$
$$\frac{dx}{d \theta} = sec \theta tan \theta$$
$$dx = sec \theta tan \theta d\theta$$

now we substitute:

$$\int \sqrt{x^2 - 1} dx$$
$$= \int \sqrt{sec^2 \theta - 1} sec \theta tan \theta d \theta$$

since $$sec^2 \theta - 1 = tan^2 \theta$$

$$= \int \sqrt{sec^2 \theta - 1} sec \theta tan \theta d \theta = \int \sqrt{tan^2 \theta} sec \theta tan \theta d \theta$$

$$= \int tan^2 \theta sec \theta d \theta$$

this is where I am stuck. A hint would be appreciated. Thanks in advance

Regards,

Express tangent in terms of secant, separate the two integrals. One will be trivial, and If i remember correctly the other can be done with a substitution.

whozum said:
Express tangent in terms of secant, separate the two integrals. One will be trivial, and If i remember correctly the other can be done with a substitution.

thanks, I got it, but I had to use Integration by parts like 5 times to get $$\int sec^3 \theta d\theta$$

Regards,

## 1. What is trigonometric substitution and why is it used in integrals?

Trigonometric substitution is a method used to solve integrals that involve expressions containing trigonometric functions. It involves substituting trigonometric identities for algebraic expressions in order to simplify the integral and make it easier to solve.

## 2. How do you know when to use trigonometric substitution in an integral?

Trigonometric substitution is typically used when the integral contains an expression of the form √(a² - x²) or √(a² + x²), where a is a constant. These can be simplified using trigonometric identities, making the integral easier to solve.

## 3. What are the steps for using trigonometric substitution in an integral?

The steps for using trigonometric substitution are as follows:

1. Identify the trigonometric function to use based on the form of the integral.
2. Choose an appropriate trigonometric substitution, such as x = a sin θ or x = a tan θ.
3. Substitute the chosen expression for x in the integral.
4. Simplify the integral using trigonometric identities.
5. Integrate the simplified expression.
6. Substitute back for θ to get the final solution.

## 4. Are there any common mistakes to watch out for when using trigonometric substitution?

Yes, there are a few common mistakes to watch out for when using trigonometric substitution. Some of these include:

• Forgetting to substitute back for θ in the final solution.
• Using the wrong trigonometric function or identity.
• Forgetting to include the constant of integration.
• Making algebraic errors when simplifying the integral.

## 5. Can trigonometric substitution be used for all integrals?

No, trigonometric substitution is only used for integrals that involve expressions containing trigonometric functions. It cannot be used for integrals that do not contain these types of expressions, such as polynomial or exponential functions.

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