Integration is the inverse process of derivative. The real problem of Intergal calculus is, given the differential coefficient of a function, to discover the function itself.

Integral Formula :             $\int_{a}^{b} f(x) dx$

Here the sign $\int$ is called as Integral. The symbol dx indicates that the integration is to be performed with respect to the variable x.

The symbol $\int$ and dx, separately are meaningless. These 2 symbols may be regarded like a pair of brackets in which the function is to be integrated.
This table gives the information of the Fundamental integral Formulas which follow immedietely from the standard differentiation formulas.

  $\frac{\mathrm{d} }{\mathrm{d} x} (x^{n+1})=(n+1) x^{n}$    $\int x^{n} dx =\frac{x^{n+1}}{n+1} + c, n\neq -1$ 
 2.  $\frac{\mathrm{d} }{\mathrm{d} x}(sin x) = cos x$  $\int cos x\ dx = sin x + c$
 3.  $\frac{\mathrm{d} }{\mathrm{d} x}(cos x) = -sin x$  $\int sin x\ dx = -cos x + c$
 4.   $\frac{\mathrm{d} }{\mathrm{d} x}(tan x) = sec^{2} x$  $\int sec^{2} x\ dx = tan x + c$