evaluate. ∫(14x^5/7 - 13x^3/5) dx ∫(14x^5/7 - 13x^3/5) dx = (type an exact answer.)

Explanation:

Step1: Apply sum - difference rule of integration

$\int(14x^{5/7}-13x^{3/5})dx=\int14x^{5/7}dx-\int13x^{3/5}dx$

Step2: Use power - rule for integration $\int x^n dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$)

For $\int14x^{5/7}dx$, we have $14\int x^{5/7}dx=14\times\frac{x^{\frac{5}{7}+1}}{\frac{5}{7}+1}=14\times\frac{x^{\frac{12}{7}}}{\frac{12}{7}}=\frac{49}{6}x^{\frac{12}{7}}$ For $\int13x^{3/5}dx$, we have $13\int x^{3/5}dx=13\times\frac{x^{\frac{3}{5}+1}}{\frac{3}{5}+1}=13\times\frac{x^{\frac{8}{5}}}{\frac{8}{5}}=\frac{65}{8}x^{\frac{8}{5}}$

Step3: Combine the results

$\int(14x^{5/7}-13x^{3/5})dx=\frac{49}{6}x^{\frac{12}{7}}-\frac{65}{8}x^{\frac{8}{5}}+C$

Answer:

$\frac{49}{6}x^{\frac{12}{7}}-\frac{65}{8}x^{\frac{8}{5}}+C$