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?
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.