Question

question 2 · 1 point
if (f(x)=-\frac{x^{2}}{3}) is an antiderivative of (f(x)), find (int(-3f(x)-2x^{3})dx).
provide your answer below:

Explanation:

Step1: Recall antiderivative property

If $F(x)$ is an antiderivative of $f(x)$, then $\int f(x)dx = F(x)+C$ and $\int af(x)dx=aF(x)+C$ for a constant $a$.

Step2: Split the integral

We know that $\int(- 3f(x)-2x^{3})dx=\int - 3f(x)dx-\int2x^{3}dx$.
Since $F(x)$ is an antiderivative of $f(x)$, $\int - 3f(x)dx=-3F(x)$.
For $\int2x^{3}dx$, using the power - rule $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n
eq - 1$), when $n = 3$, we have $\int2x^{3}dx=2\times\frac{x^{4}}{4}=\frac{1}{2}x^{4}$.

Step3: Combine the results

$\int(-3f(x)-2x^{3})dx=-3F(x)-\frac{1}{2}x^{4}+C$.

Answer:

$-3F(x)-\frac{1}{2}x^{4}+C$