Question

Evaluate $\int \frac{10n^9+{10}^{n}ln10}{\sqrt{n^{10}+{10}^n+{10}^{10}}}dn$.

Answer 1

$\int \frac{10n^9+{10}^{n}ln10dn}{\sqrt{n^{10}+{10}^n+{10}^{10}}}$

$={10}^{10}+n^{10}+{10}^n=t$

$10n^9+{10}^{n}ln10dn=dt$

$=\int \frac{dt}{\sqrt{t}}$

$=\frac{t^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}=2t^{\frac{1}{2}}+c$

$=2({10}^n+n^{10}+{10}^{10})+C$.