Question

unit 5 — challenge 5.3: antiderivatives
use the fundamental theorem of calculus to evaluate \\(\int_{-2}^{1} 5x^4 dx\\).

a.) \\(\int_{-2}^{1} 5x^4 dx = -31\\)

b.) \\(\int_{-2}^{1} 5x^4 dx = -75\\)

c.) \\(\int_{-2}^{1} 5x^4 dx = 33\\)

d.) \\(\int_{-2}^{1} 5x^4 dx = -33\\)

Explanation:

Step1: Find the antiderivative

The antiderivative of $x^n$ is $\frac{x^{n+1}}{n+1}$, so for $5x^4$:
$\int 5x^4 dx = 5 \cdot \frac{x^{5}}{5} + C = x^5 + C$

Step2: Apply Fundamental Theorem of Calculus

Evaluate $x^5$ at bounds $1$ and $-2$, subtract:
$F(1) - F(-2) = (1)^5 - (-2)^5$

Step3: Compute the final value

Calculate each term and simplify:
$1 - (-32) = 1 + 32 = 33$

Answer:

c.) $\int_{-2}^{1} 5x^4 dx = 33$