How to Evaluate Integrals?

Integration and differentiation are concepts of Calculus and integrals are part of Integration. Here we will see how to evaluate integrals. Before learning methods of evaluating integrals we must study integrals. Whenever we are asked to find area under any curve then integral is the best method to calculate it because with help of integrals we can calculate as much area as we want by applying two limits, one limit will be the initial Point from which we will be calculating area and other limit will be the final point.
There are lot of methods using which we can calculate the integral but here we will be discussing some important methods only.
There are many methods of evaluating integrals; some of them are given below,
Integration by substitution
Integration by Parts
Integration using U- substitution
Integration difference formula
These are the four most important methods used in integration, now we will see how we can integrate a function using Integration by substitution, here we are using substitution method because it will cover some other methods as well,
Example 1: Evaluate the given integral by substitution, f(x)= ∫ sin (x + 5) dx?
Solution:
Let x + 5 = t,
For getting the value of 'dx' we will differentiate it,
=> (1) dx + 0 = dt,
=> dx = dt,
Now we will substitute the value,
Therefore above equation can be written as,
=> f(x) = ∫ sin(t) dt,
=> f(x) = (-cos (t)) + c,
Now we will substitute the value of 't' to get the desired answer.
=> f (x) = - (cos (x + 5)) + c,
This is the required answer for the given problem.