Question

Any can integrate this problem​

Answer 1

$\rm \large \longrightarrow \displaystyle \int \: \rm \dfrac{ \rm 10 {x}^{9} + {10}^{x} \: log(10) }{ \rm{x}^{10} + {10}^{x} } \: dx \\ \\ \\ \rm \: let \: \: \: \rm{x}^{10} + {10}^{x} = t \\ \\ \rm \:( 10 {x}^{9} + {10}^{x} \: log10 \: ) dx= dt \\ \\ \\ \rm \large \longrightarrow \displaystyle \int \: \rm \dfrac{ dt}{ t } \\ \\ \\ \rm \large \longrightarrow \displaystyle \: \rm \: log(t) + c\\ \\ \\ \rm \large \longrightarrow \displaystyle \: \rm \: log({x}^{10} + {10}^{x}) + c$

Answer :-

log(x¹⁰+ 10ˣ) + c

Let's see how we differentiated 10ˣ while solving the above question :-

$\rm let \: \: y = {10}^{x} \\ \\ \rm \rightarrow log(y) = log({10}^{x}) \\ \\ \rightarrow \rm \: log(y) =x \: log({10})\\ \\ \rightarrow \rm \: \frac{1}{b} \times \frac{dy}{dx} = log({10})\\ \\ \rightarrow \rm \: \frac{dy}{dx} =b \times log({10})\\ \\ \rightarrow \rm \: \frac{dy}{dx} = {10}^{x} \: log({10})$