Question

$\int \frac{{\left(2^x+3^x\right)}^2}{6^x}dx$

Answer 1

$$\begin{gathered} \int \frac{{\left(2^x+3^x\right)}^2}{6^x}dx \\ =\int \left[\frac{{\left(2^x\right)}^2+{\left(3^x\right)}^2+2·2^x·3^x}{6^x}\right]dx \\ =\int \left(\frac{{\left(2^x\right)}^2}{2^x·3^x}+\frac{{\left(3^x\right)}^2}{2^x·3^x}+\frac{2·2^x·3^x}{2^x·3^x}\right)dx \\ \Rightarrow \int \left[{\left(\frac{2}{3}\right)}^x+{\left(\frac{3}{2}\right)}^x+2\right]dx \\ \Rightarrow \frac{{\left({\displaystyle \frac{2}{3}}\right)}^x}{\ln \left({\displaystyle \frac{2}{3}}\right)}+\frac{{\left({\displaystyle \frac{3}{2}}\right)}^x}{\ln {\displaystyle \frac{3}{2}}}+2x+C\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\left(\because \int a^{x}dx\mathit{=}\frac{a^x}{\ln a}\right) \end{gathered}$$