`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 about 1 month ago

Solution by Steps

step 1

We need to compute the integral of the function $g(x) = 5x^4 - x^2 + 8$ from $x = 0$ to $x = 3$

step 2

The integral can be expressed as $\int_0^3 (5x^4 - x^2 + 8) \, dx$

step 3

Evaluating the integral, we find: $\int (5x^4) \, dx = x^5$, $\int (-x^2) \, dx = -\frac{x^3}{3}$, and $\int 8 \, dx = 8x$. Thus, the antiderivative is $x^5 - \frac{x^3}{3} + 8x + C$

step 4

Now, we evaluate this from 0 to 3: $[3^5 - \frac{3^3}{3} + 8(3)] - [0 - 0 + 0] = 243 - 9 + 24 = 258$

Answer

258

Key Concept

The Fundamental Theorem of Calculus allows us to evaluate definite integrals by finding the antiderivative of the function.

Explanation

The integral of the polynomial function was computed by finding its antiderivative and then evaluating it at the bounds of integration, resulting in a final value of 258.

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