Question

intergating (6x^6/7-7x5/7)

Answer 1

\[\int\left(\frac{6x^6}{7}-\frac{7x^5}{7}\right)~dx\]
or
\[\int\left(6x^{6/7}-7x^{5/7}\right)~dx~~?\]

Answer 2

its the second on

Answer 3

Apply the power rule for integration:
\[\int x^n~dx=\frac{x^{n+1}}{n+1}+C~~\text{for }n\not=-1\]
In this case, \(n_1=\dfrac{6}{7}\) and \(n_2=\dfrac{5}{7}\), so
\[\begin{align*}\int\left(6x^{6/7}-7x^{5/7}\right)~dx&=6\frac{x^{(6+7)/7}}{\frac{6+7}{7}}-7\frac{x^{(5+7)/7}}{\frac{5+7}{7}}+C\\\\&=6\frac{x^{13/7}}{\frac{13}{7}}-7\frac{x^{12/7}}{\frac{12}{7}}+C\\\\
&=\frac{6\cdot7}{13}x^{13/7}-\frac{7\cdot7}{12}x^{12/7}+C\\\\
&=\frac{42}{13}x^{13/7}-\frac{49}{12}x^{12/7}+C
\end{align*}\]