If the primitive function of an integrand can be found it is always best to take advantage of the fundamental theorem of calculus.
In order to be able to determine integrals whose indefinte integrals(primitive functions) cannot be found immediately some of the following motions can be useful:
- variablesubstitution: This method can be developed by integrating the chain rule:
- d/dx(f(g(x)) = f´(g(x)) g'(x) This gives:
∫f'(g(x))g'(x)dx = f((g(x)) + C
Then perform the substitution u = g(x). If we differentiate this we get du = g'(x) dx.
Substitution in the integral above yields the following:
∫ f(g(x)) g'(x) dx = ∫f ‘(u) du = f(u) + C
substituting back to g(x) and we have the answer: f((g(x)) + C.
Example: Determine ∫x/(x2+1)dx
This integral can be dealt with tby using the substitution
u = x2 + 1.
Then du = 2x dx → x dx = du/2.
Substitution transforms the integral to: ∫du/(2u) = ln|u| + C.
√Substituting back gives us the answer: 0,5·ln|x2 +1| + C = ln√(x2 + 1).