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Evaluate the following integrals:

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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