**Introduction (Integration)**

Integration is the reverse process of differentiation. It is also called anti-differentiation.

Integral 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),

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 results-

- If dy = f’(x) dx, then
- If dy = a [f’(x) dx] where a is constant, then-

The integration of x^n will be as follows-

**Example: Find the integral of-**

Sol:

We know that-

hence,

**Example: Find the integral**

Sol.

We know that-

then

**Integration by substitution**

**Example: Evaluate the following integral-**

Sol:

Let us suppose,

or

therefore, Substituting –

**Logarithmic function **

**Example: Evaluate the following integral-**

**Sol:**

**Let us suppose-**

**Integration of exponential function**

**Example: Evaluate-**

**Sol:**

**Let**

Now substituting-

## Method for **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-

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