find the antiderivative $f(x)$ with $f(x)=sqrt{x^{11}}$ and $f(0) = 0$.\n$f(x)=\n$\nis there only one…

Explanation:

Step1: Rewrite the function

Rewrite $\sqrt{x^{11}}$ as $x^{\frac{11}{2}}$.

Step2: Use antiderivative formula

The antiderivative of $x^n$ is $\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$). For $n=\frac{11}{2}$, we have $\int x^{\frac{11}{2}}dx=\frac{x^{\frac{11}{2}+1}}{\frac{11}{2}+1}+C=\frac{x^{\frac{13}{2}}}{\frac{13}{2}}+C=\frac{2}{13}x^{\frac{13}{2}}+C$.

Step3: Determine the constant C

Given $F(0) = 0$, substitute $x = 0$ and $F(0)$ into $F(x)=\frac{2}{13}x^{\frac{13}{2}}+C$. We get $0=\frac{2}{13}(0)^{\frac{13}{2}}+C$, so $C = 0$.

Answer:

$F(x)=\frac{2}{13}x^{\frac{13}{2}}$ Choose one: Yes