Definite Integral of x² + 2x from 0 to 2

Published on October 7, 2024

Calculus

Learn how to solve the definite integral ∫₀² (x² + 2x) dx, step-by-step, and discover the answer: 20/3.

Question

Evaluate the definite integral: ∫₀² (x² + 2x) dx

Let’s break down how to solve this definite integral step-by-step.

1. Find the Antiderivative

We need to find the antiderivative of the function inside the integral, (x² + 2x).
Remember that the antiderivative of x^n is (x^(n+1))/(n+1).
- The antiderivative of x² is (x³)/3.
- The antiderivative of 2x (which is 2x¹) is (2x²)/2 = x².
Therefore, the antiderivative of (x² + 2x) is (x³/3) + x².

2. Evaluate the Antiderivative at the Limits of Integration

Our limits of integration are 0 and 2. We need to evaluate the antiderivative at these points.
- Upper Limit (x = 2): [(2³)/3] + 2² = (8/3) + 4 = 20/3
- Lower Limit (x = 0): [(0³)/3] + 0² = 0

3. Subtract the Lower Limit Value from the Upper Limit Value

Subtract the value of the antiderivative at the lower limit from the value at the upper limit:

(20/3) - 0 = 20/3

Answer:

The definite integral ∫₀² (x² + 2x) dx = 20/3