Integration

Integration is one of the main branches of calculus, counterpart to differential calculus. Single-variable integration is concerned with area.

The functions within the operation are called the integrand. Integrals that are bounded are definite integrals. Unbounded integrals are indefinite integrals, or anti-derivatives.

Our intuitive understanding of how integration works begins with Riemann sums. The idea is that we approximate with some small change in quantity, sum them up, and progressively improve our approximation by taking an infinitely large amount of quantities within the bounds.

Key concepts

• Riemann sum
• Fundamental theorem of calculus
• Integration techniques
  • Change of variables
    • U-substitution
    • Trigonometric substitution
  • Integration by parts
  • Partial fraction decomposition
• Improper integral
• Double integral
• Triple integral
• Line integral
• Surface integral

Applications

• Integrator
• Volume of revolution
• Work
• Arc length

Complex integration

• Contour integral
• Indentation of contours
• Principal value integral
• Trigonometric integral

Table of integrals

$$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C,n\ne -1$$ $$\int \frac{1}{x}dx=\ln |x|+C$$ $$\int \sin xdx=-\cos x+C$$ $$\int \cos xdx=\sin x+C$$ $$\int e^{x}dx=e^x+C$$ $$\int a^{x}dx=\frac{1}{\ln a}{a}^x+C$$

A useful result of complex numbers is that if $f(t)=g(t)+ih(t)$, then:

$${\int }_a^{b}f(t)dt={\int }_a^{b}g(t)+i{\int }_a^{b}h(t)$$

Trigonometric integrals

$$\int \sec xdx=\ln |\sec x+\tan x|+C$$ $$\int {\sec }^{3}xdx=\frac{1}{2}(\sec x\tan x+\ln |\sec x+\tan x|)+C$$ $$\int \csc xdx=-\ln |\cot x+\csc x|+C$$ $$\int \tan xdx=-\ln |\cos x|+C$$ $$\int \cot xdx=\ln |\sin x|+C$$

For integrands in the form ${\sin }^{n}x$ or ${\cos }^{n}x$, where $\{n\in {\mathbb{Z}}^{+}\cap n\le 3\}$, see multiple angle identities.