Integration By Parts Formula

Integration by parts is a technique used to integrate the product of two or more functions where the integration cannot be performed using normal techniques. The formula for integration by parts is:

∫ f(x)·g(x)·dx = f(x) ∫ g(x)·dx – ∫(f'(x) ∫g(x)·dx)·dx + C

Here, f(x) and g(x) are the two functions to be integrated. Among the two functions, the first function f(x) is chosen such that its derivative formula exists, and the second function g(x) is selected such that an integral formula of that function exists.

Derivation of the Formula

The formula for integration by parts can be derived using the product rule of differentiation. The product rule states that the derivative of the product of two functions is equal to the first function times the derivative of the second function plus the second function times the derivative of the first function. Mathematically, it can be written as:

(d/dx) [f(x)·g(x)] = f(x)·g'(x) + g(x)·f'(x)

Rearranging this equation, we get:

f(x)·g'(x) = (d/dx) [f(x)·g(x)] – g(x)·f'(x)

Integrating both sides of the equation with respect to x, we get:

∫ f(x)·g'(x)·dx = ∫ [(d/dx) [f(x)·g(x)]]·dx – ∫ g(x)·f'(x)·dx

Simplifying the first integral on the right-hand side using the fundamental theorem of calculus, we get:

∫ f(x)·g'(x)·dx = f(x)·g(x) – ∫ g(x)·f'(x)·dx

Rearranging this equation, we get the formula for integration by parts:

∫ f(x)·g(x)·dx = f(x) ∫ g(x)·dx – ∫(f'(x) ∫g(x)·dx)·dx + C

Examples

Let’s take an example to understand how to use the integration by parts formula:

Example: ∫ x·sin(x)·dx

Choose u and v:

• u = x
• v = -cos(x)

Differentiate u:

• u’ = 1

Integrate v:

• ∫ -cos(x)·dx = sin(x)

Now put it together:

• ∫ x·sin(x)·dx = -x·cos(x) – ∫ sin(x)·dx
• ∫ x·sin(x)·dx = -x·cos(x) – cos(x) + C

In conclusion, integration by parts is a useful technique for integrating the product of two or more functions. The formula for integration by parts can be derived using the product rule of differentiation. By choosing the right functions for u and v, we can simplify the integration process and solve complex integrals.