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:

$\int u . dv = u.v - \int v .du$

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

Multiplying both side of product rule formula by $dx$

$d(u.v) = u \times dv + v \times du$

or ,

$u. dv = d(u.v) - v.du$

Now integrating both side of above equation we get:

$\int u .dv = u.v - \int v . du$

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 :

$\int x \ln x .dx$

Let , $u = \ln x$ and $dv = x.dx$

then , $\frac{du}{dx} = \frac{1}{x}$ and $\int dv = \int ( x . dx)$

So, $du = \frac{dx}{x}$ and $v = \frac{x^2}{2}$

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

$\int x \ln x dx = \dfrac{x^2 . \ln x}{2} - \int \frac{x^2}{2} \times \frac{dx}{x}$

$= \dfrac{x^2. \ln x}{2} - \dfrac{x^2}{4} + c$

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