Question

Find the integral of $\frac{9x^8+9^x\ln 9}{x^9+9^x}$ with respect to $x$.

Answer 1

$I=\int \frac{9x^8+9^x\ln 9}{x^9+9^x}dx$

Put $x^9+9^x=t$
$\Rightarrow (9x^8+9^x\ln 9)dx=dt$
$\therefore I=\int \frac{dt}{t}$
$=\ln t+C$
$=\ln (x^9+9^x)+C$