If u and v are functions of x, then the Bernoulliâ€™s rule is âˆ«udv = uv âˆ’ u â€²v1 + u â€²â€²v2 - ......

Bernoulliâ€™s formula for Integration by Parts

If u and v are functions of x, then the Bernoulliâ€™s rule is

âˆ«udv = uv âˆ’ u â€²v1 + u â€²â€²v2 - ......

where u â€², u â€²â€², uâ€²â€²â€²,... are successive derivatives of u

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

Bernoulliâ€™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 Bernoulliâ€™s formula helps to find the solution easily.

Integrate the following with respect to x.

(i) x2 e5x (ii) x3 cos x (iii) x3eâˆ’ x

Integrate the following with respect to x:

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11th Mathematics : UNIT 11 : Integral Calculus : Bernoulliâ€™s formula for Integration by Parts |

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