Trig substitution is a technique used to simplify and solve integrals involving square roots of quadratic expressions. It is typically used when the integral contains a term in the form of √(a^2 − x^2), where a is a constant.
To perform trig substitution, you first identify the appropriate trigonometric substitution by comparing the square root expression to the standard trigonometric identities. Then, you substitute the trigonometric function for the variable, making additional substitutions as needed to simplify the integral. Finally, you solve the integral using trigonometric identities and integration techniques.
The most common types of trig substitutions are u-substitution, reverse u-substitution, and double angle substitution. U-substitution involves substituting √(a^2 − x^2) with a trigonometric function such as sinθ or cosθ. Reverse u-substitution involves substituting a trigonometric function for a variable in the form of √(x^2 − a^2). Double angle substitution involves using double angle identities to rewrite the integral in a form suitable for trig substitution.
Partial fractions decomposition is a method used to break down a rational function into simpler fractions. It is typically used to simplify integrals involving rational functions. The decomposition involves finding the appropriate coefficients for each term in the fraction by equating the numerator and denominator on both sides of the equation.
Partial fractions decomposition is used when the integral contains a rational function that cannot be easily integrated using other techniques. It is also used to solve systems of linear equations and to find Laplace transforms in mathematics and in engineering applications.