Question

Find value of $k$
$\int \frac{x^{5/2}}{\sqrt{1+x^7}}dx=\frac{1}{k}log(x^k+\sqrt{1+x^7})+C$

Answer 1

Substitute $x^{7/2}=t$
$\therefore \frac{7}{2}{x}^{5/2}dx=dt$
$\therefore I=\int \frac{2}{7}\frac{dt}{\sqrt{1+t^2}}=\frac{2}{7}log(t+\sqrt{1+t^2})+C$
$=\frac{2}{7}log(x^{7/2}+\sqrt{1+x^7})+C$