Question

The value of ${\int }_1^{\sqrt[7]{72}}\frac{1}{x(2x^7+1)}dx$ is

• A. $\log \left(6/5\right)$
• B. $6\log \left(6/5\right)$
• C. $\left(1/7\right)\log \left(6/5\right)$
• D. $\left(1/12\right)\log \left(6/5\right)$

Answer 1

Answer: (C)

$\left(1/7\right)\log \left(6/5\right)$

${\int }_1^{21/7}\frac{x^{6}dx}{x^7(2x^7+1)}=I$

$x^7=\mu \Rightarrow 7x^{6}dx=du$

$I=\frac{1}{7}{\int }_1^2\frac{du}{\mu (2\mu +1)}$

$\frac{1}{7}=\frac{1}{7}{\int }_1^2(\frac{1}{\mu }-\frac{2}{2\mu +1})du$

$=\frac{1}{7}[ln\mu -ln(2\mu +1){]}_1^2$

$=\frac{1}{7}(ln\frac{2}{5}-ln\frac{1}{3})$

$=\frac{1}{7}ln\left(\frac{6}{5}\right)$

Option $C$