Integration of a Basic Function

Integration is essentially the reverse process of differentiation. It calculates the total sum of many infinitesimal quantities. These are the basics points on integration of a basic function:

The Power Rule: If n ≠ -1, then the integral of x^n with respect to x is 1/(n+1) * x^(n+1).
Integrating a constant multiplied by a function: The integral of a constant times a function is the constant times the integral of the function.
Integral of a sum: The integral of a sum or difference of functions is the sum or difference of their integrals.
The integral of a constant, k, is kx.

Techniques of Integration

Integrating using substitution: A powerful technique that makes the function simpler to integrate.
Integrating using parts: This is best for products of two functions, especially when one is easily differentiable and the other is easily integrable.
Partial fractions: This technique makes it easier to integrate rational functions (fractions where the numerator and denominator are polynomials).

Applications of Integration

Finding Area Under a Curve: This is the most common application of integration.
Solving Differential Equations: Integration is used to find the functions that satisfy differential equations.
Volume of revolution: Integration can calculate the volume of a 3-dimensional object created by revolving a 2-dimensional shape around an axis.
Work done: Physics often uses integration to calculate the work done by a variable force.
Probability density function: In probability theory, the area under the probability density function is calculated using integration to find the probability of an event.