Trigonometric substitution is a technique for evaluating integrals by substituting trigonometric functions for algebraic expressions. This method is particularly useful for integrals involving square roots of quadratic expressions.
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Trigonometric substitutions often involve using the identities $x = a \sin(\theta)$, $x = a \tan(\theta)$, or $x = a \sec(\theta)$ to simplify the integral.
The choice of substitution depends on the form of the quadratic expression under the square root: $a^2 - x^2$, $a^2 + x^2$, or $x^2 - a^2$.
After substitution, the integral typically simplifies into a trigonometric integral which can be evaluated using standard methods.
Don't forget to change the limits of integration if you are working with definite integrals when making substitutions.
Always convert back to the original variable after integrating and simplify your final answer.
A fundamental concept in calculus that represents the area under a curve or, more generally, accumulation of quantities.
Trig Identities: Equations involving trigonometric functions that are true for all values within their domains. Examples include $\sin^2(x) + \cos^2(x) = 1$.