Integral

It can be seen as the area under the curve. Calculating the integral of a function can be considered as calculating the "antiderivative". Given the derivative of a function we have infinite number of "antiderivative" functions since moving a function curve vertically does not affect its derivative function. For convenience, we can always think an integral function as the one which has its $f(0)=0$ by moving it vertically.

Rules

Integration by substitution: $\int _{a}^{b}f(\varphi (x))\varphi '(x)\,dx=\int _{\varphi (a)}^{\varphi (b)}f(u)\,du$

Power rule: $\int x^n = \frac{x^{n+1}}{n+1} + C$