Integration by Parts Integration by parts is a technique used to solve integrals involving products of two functions. It allows us to rewrite the integral a...
Integration by Parts
Integration by parts is a technique used to solve integrals involving products of two functions. It allows us to rewrite the integral as the sum of two simpler integrals, making it easier to evaluate.
Step 1: Choose two functions:
Step 2: Apply the formula:
∫(fg)'dx = f'(x)∫gdx + f(x)∫g'(x)dx
Step 3: Evaluate the integrals:
∫f'(x)dx = f(x)dx
∫g'(x)dx = g(x)dx
Step 4: Combine the integrals:
∫(fg)'dx = f(x)dx - ∫f(x)dx + ∫g(x)dx
Example:
Evaluate the following integral using integration by parts:
∫x(x+1)dx
Solution:
Choose f(x) = x and g(x) = x+1. Then:
∫x(x+1)dx = ∫x^2dx - ∫xdxdx
Evaluating the integrals, we get:
∫x^2dx = (1/3)x^3
Therefore:
∫x(x+1)dx = (1/3)x^3 - x + C
where C is the constant of integration