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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