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# Bernoulli‚Äôs formula for Integration by Parts

If u and v are functions of x, then the Bernoulli‚Äôs rule is ‚ąęudv = uv ‚ąí u ‚Ä≤v1 + u ‚Ä≤‚Ä≤v2 - ......

Bernoulli‚Äôs formula for Integration by Parts

If u and v are functions of x, then the Bernoulli‚Äôs rule is

‚ąęudv = uv ‚ąí u ‚Ä≤v1 + u ‚Ä≤‚Ä≤v2 - ......

where u ‚Ä≤, u ‚Ä≤‚Ä≤, u‚Ä≤‚Ä≤‚Ä≤,... are successive derivatives of u

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

Bernoulli‚Äôs 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‚Äôs 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‚ąí x

### EXERCISE 11.7

Integrate the following with respect to x: Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
11th Mathematics : UNIT 11 : Integral Calculus : Bernoulli‚Äôs formula for Integration by Parts |