# Integration, Techniques of Integration, Integration of Function of the Type f‵ (x)/f (x), Integration by Substitution

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## Techniques of Integration

This method consists of expressing in terms of another variable so that the resultant function can be integrated using one of the standard results discussed in the previous lesson. First, we will consider the functions of the type where is a standard function.

Example:

Evaluate:

Solution:

Put .

Then or

(Here the integration factor will be replaced by .)

## Integration of Function of the Type F‵(X)/F(X)

To evaluate , we put Then

Integral of a function, whose numerator is derivative of the denominator, is equal to the logarithm of the denominator.

Example:

Evaluate:

Solution:

Now is the derivative of .

By applying the above result, we have

## Integration by Substitution

Solution:

( is derivative of )

or

can not be integrated as such because by itself is not derivative of any function. But this is not the case with and . Now can be written as

Put .

Then

Example:

Evaluate:

Solution:

Put