Bernoulli’s Formula

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 x^n−1 + + a_n ) 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 ^π∫₀ 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 : ¹âˆ«₀ e^−2x (1 + x − 2x³ ) dx .

Solution

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

I = ¹âˆ«₀ e^−2x (1+ x − 2x³ ) dx

Example 9.33

Evaluate : ^2π∫₀ 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 : ¹âˆ«_−1 e^{− λx} (1 − x² ) dx .

Solution

Taking u = 1− x² and v = e^{−λ x} , and applying the Bernoulli’s formula, we get