Bernoulli’s formula for Integration by Parts

Bernoulli’s formula for Integration by Parts

If u and v are functions of x, then the Bernoulli’s rule is

∫udv = uv − u ′v1 + u ′′v2 - ......

where u ′, u ′′, u′′′,... are successive derivatives of u

and v, v1 , v2 , v3 , are successive integrals of dv

Bernoulli’s formula is advantageously applied when u = xn ( n is a positive integer)

For the following problems we have to apply the integration by parts two or more times to find the solution. In this case Bernoulli’s formula helps to find the solution easily.

Example 11.35

Integrate the following with respect to x.

(i) x2 e5x (ii) x3 cos x (iii)  x3e− x

EXERCISE 11.7

Integrate the following with respect to x: