## Integration by parts formula

### The integration by parts formula is a rule used to find the integral of a product of two functions

The integration by parts formula is a rule used to find the integral of a product of two functions. It is derived from the product rule of differentiation.

The formula states:

∫ u dv = u v – ∫ v du

where:

– ∫ represents the integral sign

– u and v are functions of the variable of integration

– du represents the derivative of the function u with respect to the variable of integration

– dv represents the derivative of the function v with respect to the variable of integration

To understand how to apply the integration by parts formula, let’s consider an example:

Example:

Find the integral of x * sin(x) dx

Solution:

In this problem, we will assign u = x and dv = sin(x) dx.

Now, we need to find du and v by taking the derivatives and integrals respectively.

Taking the derivative of u with respect to x, we have du = dx.

Integrating dv, we have ∫ dv = ∫ sin(x) dx = -cos(x) (using the integral of sin(x) = -cos(x)).

Using the integration by parts formula:

∫ x * sin(x) dx = u v – ∫ v du

= x * (-cos(x)) – ∫ (-cos(x)) dx

= -x * cos(x) + ∫ cos(x) dx

= -x * cos(x) + sin(x) + C

where C is the constant of integration.

So, the integral of x * sin(x) dx is equal to -x * cos(x) + sin(x) + C.

The integration by parts formula is a powerful tool that can be used to integrate a wide range of functions, especially when the original function contains a product of two functions that are difficult to integrate separately.

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