Definite Integral of a Power

\int_{a}^{b} x^{n} d x = [\frac{x ^{n + 1}}{n + 1}]_{a}^{b} (n \neq = - 1)

The definite integral of x^n from a to b is the net signed area. Evaluate the antiderivative F(b) − F(a). For n = −1 the antiderivative is ln|x|. By FTC, d/dx ∫_a^x f(t) dt = f(x).

$a$: Lower limit
$b$: Upper limit
$n$: Exponent, n ≠ −1
$x^{n + 1} / (n + 1)$: Antiderivative

Step-by-step solved example

Evaluate ∫ from 0 to 2 of x² dx.

Definite integral is the signed area under the curve (same idea as U=∫σ dε).

1. Antiderivative
∫ x² dx = x³/3.
$x^{3} /3$
2. Evaluate bounds
[x³/3]_0^2 = 8/3 − 0 = 8/3 ≈ 2.667.
$8/3$

Answer: 8/3

10 practice questions

0/10 correct

1.∫_0^1 x dx equals

2.∫_0^2 x² dx equals

3.∫_1^2 3x² dx equals

4.∫_a^a f(x) dx equals

5.∫_0^3 2x dx equals

6.∫_0^1 x^n dx for n > −1 equals

7.The power antiderivative fails at n = −1 because

8.∫_2^4 x dx equals

9.By FTC, d/dx ∫_0^x t² dt equals

10.Area under y = 3 from x = 1 to 5 is