Introduction– Integration is the reverse process of differentiation. in other words It is also called anti-differentiation.
Integration calculus has its own application in economics, Engineering, Physics, Chemistry, business, commerce, etc.
The integral of a function is denoted by the sign
Let the function is y = f(x), So that its derivative is-
Then
Where c is the arbitrary constant.
For example,
A function,
Then, its derivative-
Or
Then
Here c is an arbitrary constant.
Some fundamental integrals-
Methods of integration
Simple integration-
1. When the function is an algebraic function-
Some standard form are-
The integration of x^n will be as follows-
Example: Find the integral of-
Sol.
We know that-
Then
Example: Find the integral
Sol.
We know that-
Then
Example: Evaluate-
Sol.
By substitution
Example: Evaluate the following integral-
Sol.
Let us suppose,
Then
Or
Substituting –
Logarithmic function-
Example: Evaluate the following integral-
Sol.
Let us suppose-
Now
Integral of exponential function
Example: Evaluate-
Sol.
Let,
Now substituting-
Integration of product of two functions-
Suppose we have two function say- f(x) and g(x), then
The integral of product of these two functions is-
Note-
We chose the first function as method of ILATE-
Which is-
I – Inverse trigonometric function
L – Log function
A – Algebraic function
T- Trigonometric function
E- Exponential function
Example: Evaluate-
Sol.
Here according to ILATE,
First function = log x
Second function = x^n
We know that-
Then
On solving, we get-
