Lesson 10 of 18Article22 min
Integration Techniques: Substitution & Parts
When the integrand looks like f(g(x)) g′(x), set u = g(x), du = g′(x) dx, and integrate in u. For definite integrals, change the limits to u-values — no need to return to x.
u-substitution
When the integrand looks like f(g(x)) g′(x), set u = g(x), du = g′(x) dx, and integrate in u. For definite integrals, change the limits to u-values — no need to return to x.
Integration by parts
The product rule in reverse. Choose u to become simpler when differentiated; dv to be easy to integrate.
Classic: ∫ x eˣ dx with u=x, dv=eˣ dx → x eˣ − ∫ eˣ dx = eˣ(x−1) + C.
Partial fractions (preview)
Rational functions often split into simpler pieces A/(x−a) + B/(x−b) after factoring the denominator. Integrate each term with logs or arctan forms.