The integration by parts rule: ∫ u dv = u*v - ∫ v du
This rule is used to integrate the product of two functions. The result of the integration by parts formula is obtained by differentiating one of the functions and integrating the other function.
The formula can be written as:
∫ u dv = u*v - ∫ v du
where ∫ represents the integral sign, and u and v are functions of the variable of integration. The derivative of u is represented by du, and the integral of v is represented by dv.
To use this formula, we need to identify which function to choose for u and which one for dv. Our choice should be based on the fact that the derivative of u should be simpler to integrate than the original u itself.
For example, let's evaluate the integral of x*e^x. We can choose u = x and dv = e^x dx. Then, we have:
du = dx (since the derivative of x is 1)
v = e^x (the integral of e^x is e^x)
Using the integration by parts formula gives:
∫ x*e^x dx = x*e^x - ∫ e^x dx
= x*e^x - e^x + C (where C is the constant of integration)
We can check that this solution is correct by taking its derivative:
d/dx (x*e^x - e^x + C) = e^x + x*e^x - e^x
= x*e^x
which is indeed the integrand of the original function.