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.

No.
Differentiation
Integration
 1.
  $\frac{\mathrm{d} }{\mathrm{d} x} (x^{n+1})=(n+1) x^{n}$    $\int x^{n} dx =\frac{x^{n+1}}{n+1} + c, n\neq -1$ 
 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.   $\frac{\mathrm{d} }{\mathrm{d} x}(tan x) = sec^{2} x$  $\int sec^{2} x\ dx = tan x + c$