Question

Choose the correct. answer.
$\int \left(\frac{10x^9+{10}^{x}lo{g}_{e}10}{x^{10}+{10}^x}\right)dx$ equals
$\left(a\right){10}^x+x^{10}+C\left(b\right){10}^x+x^{10}+C\left(c\right)({10}^x-x^{10}{)}^{-1}+C(d)log|{10}^x+x^{10}|+C$

Answer 1

$\int \left(\frac{10x^9+{10}^{x}lo{g}_{e}10}{x^{10}+{10}^x}\right)dx$
Let $x^{10}+{10}^x=t\Rightarrow (10x^9+{10}^{x}lo{g}_{e}10)dx=dt$
$\Rightarrow dx=\frac{dt}{10x^9+{10}^{x}lo{g}_{e}10}\therefore \int \left(\frac{10x^9+{10}^{x}lo{g}_{e}10)}{x^{10}+{10}^x}\right)dx=\int \frac{10x^9+{10}^{x}lo{g}_{e}10}{t}\times \frac{dt}{10x^9+{10}^{x}lo{g}_{e}10}=\int \frac{dt}{t}=log|t|+C=log|{10}^x+x^{10}|+C$
So, (d) is the correct option.