Methods to Find Integrals

There are several methods adopted for finding the indefinite integrals. The prominent methods are:

• Finding integrals by integration by substitution method
• Finding integrals by integration by parts
• Finding integrals by integration by partial fractions.

Finding Integrals by Substitution Method

A few integrals are found by the substitution method. If u is a function of x, then u’ = du/dx.

∫ f(u)u’ dx = ∫ f(u)du, where u = g(x).

Finding Integrals by Integration by Parts

If two functions are of the product form,  integrals are found by the method of integration by parts.

∫f(x)g(x) dx = f(x)∫ g(x) dx – ∫ (f'(x) ∫g(x) dx) dx. 

Finding Integrals by Integration by Partial Fractions

Integration of rational algebraic functions whose numerator and denominator contain positive integral powers of x with constant coefficients is done by resolving them into partial fractions.

To find ∫ f(x)/g(x) dx, decompose this improper rational function to a proper rational function and then integrate.

∫f(x)/g(x) dx = ∫ p(x)/q(x) + ∫ r(x)/s(x), where g(x) = a(x) . s(x)