**Bernoulliâ€™s
Formula**

The
evaluation of an indefinite integral of the form âˆ«*u* (*x*) *v*
( *x*)*dx* becomes very simple, when *u
*is a polynomial function of *x*
(that is, *u* ( *x*) = *a*_{0}
*x*^{n} +
*a*_{1} *x*^{n}^{âˆ’}1 + + *a*_{n} )
and *v* ( *x*) can be easily integrated* *successively. It is accomplished by a
formula called **Bernoulliâ€™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

âˆ« *uvdx* = *uv*_{(1)} âˆ’ *u* ^{(1)}*v*_{( 2)} + *u* ^{( 2)}*v*_{( 3)} âˆ’ *u* ^{(3)}*v*_{( 4)} + .

The
above result is called the **Bernoulliâ€™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
^{Ï€}âˆ«_{0} *x*^{2}
cos *nx dx* , where *n* is a positive integer.

**Solution**

Taking *u* = *x*^{2}
and *v* = cos *nx* , and applying the Bernoulliâ€™s formula, we get

** **

**Example 9.32**

Evaluate
: ^{1}âˆ«_{0} e^{âˆ’}^{2x} (1 + *x*
âˆ’
2*x*^{3} ) *dx* .

**Solution**

Taking *u* = 1 + *x*
âˆ’
2*x*^{3} and *v* = e^{âˆ’}^{2x} , and applying the Bernoulliâ€™s formula, we get

I = ^{1}âˆ«_{0} e^{âˆ’}^{2x} (1+ *x*
âˆ’
2*x*^{3} ) *dx*

** **

**Example 9.33**

Evaluate
: ^{2}^{Ï€}âˆ«_{0} *x*^{2}
sin *nx dx* , where *n* is a positive integer.

**Solution**

Taking *u* = *x*^{2}
and *v* = sin *nx* , and applying the Bernoulliâ€™s formula, we get

** **

**Example 9.34**

Evaluate
: ^{1}âˆ«_{âˆ’}_{1} e^{âˆ’ Î»}^{x} (1 âˆ’ *x*^{2}
) *dx* .

**Solution**

Taking *u* = 1âˆ’ *x*^{2}
and *v* = e^{âˆ’Î»} ^{x}
, and applying the Bernoulliâ€™s formula, we get

Tags : Applications of Integration , 12th Maths : UNIT 9 : Applications of Integration

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12th Maths : UNIT 9 : Applications of Integration : Bernoulliâ€™s Formula | Applications of Integration