Mathematics

$$\displaystyle \int\frac{10x^{9}+10^{x}\log _{e}10}{10^{x}+x^{10}}dx.$$


ANSWER

$$\displaystyle \log \left ( 10^{x}+x^{10} \right ).$$


SOLUTION
Let $$ \displaystyle I=\int  \frac { 10x^{ 9 }+10^{ x }\log _{ e } 10 }{ 10^{ x }+x^{ 10 } } dx$$
Put $$\displaystyle 10^{ x }+x^{ 10 }=t\Rightarrow \left( 10x^{ 9 }+10^{ x }\log _{ e } 10 \right) dx=dt$$
Therefore
$$ \displaystyle I=\int { \frac { dt }{ t }  } =\log { t } =\log { \left( 10^{ x }+x^{ 10 } \right)  } $$
Hence, option 'C' is correct.
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