Enter the function, variable of integration, and trigonometric substitution into the calculator to determine the result of the integral.
Trig Substitution Integral Calculation Formula
The following formula is used to calculate the integral using trigonometric substitution.
$$\int f(\sqrt{a^2 - x^2}) \, dx = \int f(a \cos(\theta)) \, a \sin(\theta) \, d\theta$$
Variables:
- $f$ is the function to be integrated
- $a$ is a constant
- $x$ is the original variable of integration
- $\theta$ is the new variable after trigonometric substitution
To calculate the integral, substitute $x = a \sin(\theta)$ or $x = a \cos(\theta)$, simplify the integral, and then integrate with respect to $\theta$.
What is Trig Substitution?
Trigonometric substitution is a technique used in calculus to evaluate integrals. This method is particularly useful for integrals involving square roots of quadratic expressions. By substituting a trigonometric function for the variable of integration, the integral can be simplified to a form that is easier to evaluate.
How to Use Trig Substitution?
The following steps outline how to use trigonometric substitution for integral calculations:
- Identify the appropriate trigonometric substitution based on the form of the integrand.
- Substitute the trigonometric expression for the variable of integration.
- Simplify the integral using trigonometric identities.
- Evaluate the resulting integral with respect to the new variable.
- Back-substitute the original variable to obtain the final result.
Example Problem:
Use the following variables as an example problem to test your knowledge.
Function: $ \sqrt{1 – x^2} $
Variable of Integration: $ x $
Trigonometric Substitution: $ x = \sin(\theta) $
FAQ