Integral calculus concerned with the theory and applications of integrals and integration. Integral may be definite or indefinite but the process of solving either a definite or an indefinite integral is called integration. Integration is the reversal of the process of differentiating. The basic rules of integration are presented here along with several examples. The rules only apply when the integrals exist.


Sum Rule for Integrals

Let $f(x)$ and $g(x)$ be any two functions. The sum rule says that the integral of the sum of two functions is the sum of their separate integrals.

$\int [f(x)+g(x)] dx$ = $\int f(x) dx + \int g(x) dx$

$\int [f(x)-g(x)] dx$ = $\int f(x) dx -\int g(x) dx$

Example: $\int [x^{4}+ 2x^{2}]dx$ = $\int x^{4}+\int 2x^{2}dx$

= $\frac{x^{5}}{5}+\frac{2}{3}x^{3}$

Multiplication Rule for Integrals

$\int ax^{n}dx$ = $a\int x^{n}dx$ for any constant 'a'.

That is any constant factor may be taken outside the integration sign.

Example: $\int 4x^{3}dx$ = $4\int x^{3}dx$

= $4$$\frac{x^{4}}{4}$+c = $x^{4}+c$