**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-