`Compute the integral of $g(x) = 5x^4 - x^2 + 8$ from $x = 0$ to $x = 3$.`

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Answer from Sia

Posted 4 months ago

Solution by Steps

step 1

To compute the integral of $g(x) = 5x^4 - x^2 + 8$ from $x = 0$ to $x = 3$, we start by finding the antiderivative of the function

step 2

The antiderivative of $5x^4$ is $\frac{5x^5}{5} = x^5$

step 3

The antiderivative of $-x^2$ is $-\frac{x^3}{3}$

step 4

The antiderivative of $8$ is $8x$

step 5

Combining these, the antiderivative of $g(x)$ is $x^5 - \frac{x^3}{3} + 8x + C$

step 6

We now evaluate this antiderivative from $0$ to $3$:

step 7

$\left[ x^5 - \frac{x^3}{3} + 8x \right]_0^3 = \left( 3^5 - \frac{3^3}{3} + 8 \cdot 3 \right) - \left( 0^5 - \frac{0^3}{3} + 8 \cdot 0 \right)$

step 8

Simplifying, we get $\left( 243 - 9 + 24 \right) - 0 = 258$

Answer

258

Key Concept

Definite Integral

Explanation

The definite integral of a function over an interval is found by evaluating the antiderivative at the endpoints of the interval and subtracting.

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