find an antiderivative for each function when c = 0. a. 8/7 * 7th root of x b. 1/(7 * 7th root of x^6) c…

Explanation:

Step1: Recall power - rule for antiderivatives

The power - rule for antiderivatives is $\int x^n dx=\frac{x^{n + 1}}{n+1}+C$, where $n\neq - 1$. First, rewrite $\frac{8}{7}\sqrt[7]{x}$ as $\frac{8}{7}x^{\frac{1}{7}}$.

Step2: Apply the power - rule

For the function $f(x)=\frac{8}{7}x^{\frac{1}{7}}$, using the power - rule $\int\frac{8}{7}x^{\frac{1}{7}}dx=\frac{8}{7}\times\frac{x^{\frac{1}{7}+1}}{\frac{1}{7}+1}$.

Step3: Simplify the expression

$\frac{8}{7}\times\frac{x^{\frac{8}{7}}}{\frac{8}{7}}=x^{\frac{8}{7}}$.

Answer:

$x^{\frac{8}{7}}$