Question:
Evaluate the following integrals:
$\int \frac{10 x^{9}+10^{x} \log _{e} 10}{10^{x}+x^{10}} d x$
Solution:
Assume $10^{x}+x^{10}=t$
$d\left(10^{x}+x^{10}\right)=d t$
$a^{x}=\log _{e} a$
$\Rightarrow 10 x^{9}+10^{x} \log _{e} 10=d t$
Put $t$ and $d t$ in given equation we get
$\Rightarrow \int \frac{d t}{t}$
$=\ln |t|+c$
But $\mathrm{t}=10^{x}+x^{10}$
$=\ln \left|10^{x}+x^{10}\right|+c$
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