Integration by Parts
Integration by Parts is a technique used in calculus to integrate the product of two functions. It is particularly useful when faced with integrals that involve products of functions that are not easily integrable using other methods.
Mathematical Formula
If ∫ u dv = uv - ∫ v du represents an integral, where u and v are differentiable functions of x, then Integration by Parts states that the integral can be evaluated using the formula:
∫ u dv = uv - ∫ v du
Here, u is chosen as a part of the integrand, and dv is the remaining part. The goal is to select u and dv in such a way that the integral on the right-hand side becomes simpler to evaluate.
How It Works
Integration by Parts is based on the product rule for differentiation. The formula allows us to transform the original integral into a new integral that might be easier to evaluate. By selecting the appropriate parts, the process of integration can be simplified.
Integration by Parts Examples
Integration by Parts is a technique used in calculus to integrate the product of two functions. It involves selecting parts of the integrand to differentiate and integrate, with the goal of simplifying the integral.
Mathematical Formula
If ∫ u dv = uv - ∫ v du represents an integral, where u and v are differentiable functions of x, then Integration by Parts states that the integral can be evaluated using the formula:
∫ u dv = uv - ∫ v du
Examples
Example 1
Compute the integral:
∫ x ln(x) dx
u = ln(x), dv = x dx Show Solution
Example 2
Calculate the integral:
∫ e^x sin(x) dx
u = e^x, dv = sin(x) dx Show Solution
Example 3
Find the integral:
∫ x^2 e^x dx
u = x^2, dv = e^x dx Show Solution