# Integration, Integration by Parts, Integral of the Form, Algebraic Function

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## Integration by Parts

In differentiation you have learnt that

or

Also you know that

Integrating. we have

If we take

become

I function integral of II function [differential coefficient of function integral of II function]

Here the important factor is the choice of I and II function in the product of two functions because either can be I or II function. For that the indicator will be part ‘’ of the result above.

The first function is to be chosen such that it reduces to a next lower term or to a constant term after subsequent differentiations.

In questions of integration like

(i) Algebraic function should be taken as the first function

(ii) If there is no algebraic function then look for a function which simplifies the product in ‘’ as above; the choice can be in order of preference like choosin first function

An inverse function

A logarithmic function

A trigonometric function

An exponential function.

The following examples will give a practice to the concept of choosing first function.

I function |
| |

1. | (being algebraic) | |

2. | (being algebraic) | |

3. | ||

4. | ||

5. | ||

6. | ||

(In single function of logarithm and inverse trigonometric we take unity as II function) | ||

7. | 1 |

Example:

Evaluate:

Solution:

In order of preference log x is to be taken as I function.

I II

## Integral of the Form

Where is the differentiation of . In such type of integration while integrating by parts the solution will be .

For example, consider

Let , then

So can be written as

Example:

Evaluate the following: (a) (b)

Solution:

(a)

(b)