UNIT 3

Calculus II

3.1 Integration: Standard forms, Methods of integration – by substitution, by parts and by use of partial fractions, definite integration, finding areas in simple cases

Standard form

Integration is the reverse process of differentiation. It is also called annti-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),

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-

Some standard form of simple integration-

If dy = f’(x) dx, then

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

If

, then

The integration of 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.

Integration 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

Integration 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 =

We know that-

Then-

On solving, we get-

References-