find an antiderivative for each function when c = 0. a. 8/7 * 7√x b. 1/(7 * 7√x^6) c. 7√x + 1/(7√x)

Explanation:

Step1: Recall power - rule for antiderivatives

The antiderivative of $x^n$ is $\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$).

Step2: Solve part a

The function is $f(x)=\frac{8}{7}x^{\frac{1}{7}}$. Using the power - rule, with $n=\frac{1}{7}$, the antiderivative $F(x)=\frac{8}{7}\times\frac{x^{\frac{1}{7}+1}}{\frac{1}{7}+1}$. Simplify: $\frac{8}{7}\times\frac{x^{\frac{8}{7}}}{\frac{8}{7}}=x^{\frac{8}{7}}$.

Step3: Solve part b

Rewrite the function $f(x)=\frac{1}{7x^{\frac{6}{7}}}=\frac{1}{7}x^{-\frac{6}{7}}$. Using the power - rule with $n =-\frac{6}{7}$, the antiderivative $F(x)=\frac{1}{7}\times\frac{x^{-\frac{6}{7}+1}}{-\frac{6}{7}+1}$. Simplify: $\frac{1}{7}\times\frac{x^{\frac{1}{7}}}{\frac{1}{7}}=x^{\frac{1}{7}}$.

Step4: Solve part c

Rewrite the function $f(x)=x^{\frac{1}{7}}+x^{-\frac{1}{7}}$. Using the power - rule for each term. The antiderivative of $x^{\frac{1}{7}}$ is $\frac{x^{\frac{1}{7}+1}}{\frac{1}{7}+1}=\frac{7}{8}x^{\frac{8}{7}}$, and the antiderivative of $x^{-\frac{1}{7}}$ is $\frac{x^{-\frac{1}{7}+1}}{-\frac{1}{7}+1}=\frac{7}{6}x^{\frac{6}{7}}$. So the antiderivative $F(x)=\frac{7}{8}x^{\frac{8}{7}}+\frac{7}{6}x^{\frac{6}{7}}$.

Answer:

a. $x^{\frac{8}{7}}$ b. $x^{\frac{1}{7}}$ c. $\frac{7}{8}x^{\frac{8}{7}}+\frac{7}{6}x^{\frac{6}{7}}$