# Integration by Parts

*Integration by Parts *is a powerful tool and one of the best , most used techniques of integration used to evaluate integrals.

The main formula used in integration by parts is:

This formulas is derived from the product rule for differentiation as:

Multiplying both side of product rule formula by

or ,

Now integrating both side of above equation we get:

This formulas converts the problem of integrating a function with respect to another ( u .dv ) into the problem of integrating second function with respect to first function ( v .du).

So this technique is helpful when the second integral is easier to integrate than the first one.

But the **integration by parts** can be successfully used only if the two functions “u” and “dv” is chosen properly.

There is no definite rule for the choice of ”u” and “dv” , But if we follow the following suggestions is it easier to integrate integrals using integration by parts :

1. Choose “dv” such that we can integrate “dv” easily and find “v”.

2. Choose “u” such that “du” is simpler than “u”.

**Example of integration by parts**:

Let us evaluate the integral :

Let , and

then , and

So, and

So now we can use integration by parts in our original integral as:

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