Calculus · Quick Reference

Integration Techniques Cheat Sheet

Integral Calculus · Lesson 14/20

In one line: beyond the basic power rule, several techniques handle more complex integrals — substitution, integration by parts, and partial fractions among the most common.

Key Ideas

1U-Substitution. Used when an integral contains a function and its derivative. Let u equal the inner function, rewrite in terms of u, integrate, then substitute back.

2Integration by Parts. Used for products of functions. Formula: Integral of u dv = uv - Integral of v du. Choose u to be the part that simplifies when differentiated.

3Partial Fractions. Used for rational functions; breaks a complex fraction into a sum of simpler fractions that are each easy to integrate.

4Trigonometric Integrals. Integrals involving sin, cos, and other trig functions often use identities to simplify before integrating.

5Choosing a Technique. Look at the structure: a function times its derivative suggests substitution; a product of unrelated functions suggests integration by parts; a fraction with factorable denominator suggests partial fractions.

Worked Examples

Find the integral of 2x*(x^2+1)^3 dx using u-substitution.

(x^2+1)^4 / 4 + C

Find the integral of x*e^x dx using integration by parts.

x*e^x - e^x + C

Find the integral of cos(x)*sin(x) dx using u-substitution.

sin^2(x) / 2 + C

Formulas

Integration by parts: Integral of u dv = uv - Integral of v du

Practice Yourself

Integrate 3x^2*(x^3+1)^2 dx using substitution (u=x^3+1).

(x^3+1)^3 / 3 + C

Integrate x*cos(x) dx using integration by parts.

x*sin(x) + cos(x) + C

Integrate 2x*e^(x^2) dx using substitution.

e^(x^2) + C