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Applications of Integration - Bernoulli’s Formula | 12th Maths : UNIT 9 : Applications of Integration

Chapter: 12th Maths : UNIT 9 : Applications of Integration

Bernoulli’s Formula

Bernoulli’s formula for integration of product of two functions is below

Bernoulli’s Formula

The evaluation of an indefinite integral of the form u (x) v ( x)dx becomes very simple, when u is a polynomial function of x (that is, u ( x) = a0 xn + a1 xn1 + + an ) and v ( x) can be easily integrated successively. It is accomplished by a formula called Bernoulli’s formula. This formula is actually an extension of the formula of integration by parts. To derive the formula, we use the following notation:


Proceeding in this way, we get

uvdx = uv(1)u (1)v( 2) + u ( 2)v( 3)u (3)v( 4) + .

The above result is called the Bernoulli’s formula for integration of product of two functions.

Note

Since u is a polynomial function of x , the successive derivative u( m) will be zero for some positive integer m and so all further derivatives will be zero only. Hence the right-hand-side of the above formula contains a finite number of terms only.

 

Example 9.31

Evaluate π0 x2 cos nx dx , where n is a positive integer.

Solution

Taking u = x2 and v = cos nx , and applying the Bernoulli’s formula, we get


 

Example 9.32

Evaluate : 10 e2x (1 + x 2x3 ) dx .

Solution

Taking u = 1 + x 2x3 and v = e2x , and applying the Bernoulli’s formula, we get

I = 10 e2x (1+ x 2x3 ) dx


 

Example 9.33

Evaluate : 2π0 x2 sin nx dx , where n is a positive integer.

Solution

Taking u = x2 and v = sin nx , and applying the Bernoulli’s formula, we get


 

Example 9.34

Evaluate : 11 e− λx (1 x2 ) dx .

Solution

Taking u = 1 x2 and v = eλ x , and applying the Bernoulli’s formula, we get



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12th Maths : UNIT 9 : Applications of Integration


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