Question

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

Answer 1

\[\int\frac{\left( 2^x + 3^x \right)^2}{6^x}dx\]
\[ = \int\left[ \frac{\left( 2^x \right)^2 + \left( 3^x \right)^2 + 2 \cdot 2^x \cdot 3^x}{6^x} \right]dx\]
\[ = \int\left( \frac{\left( 2^x \right)^2}{2^x \cdot 3^x} + \frac{\left( 3^x \right)^2}{2^x \cdot 3^x} + \frac{2 \cdot 2^x \cdot 3^x}{2^x \cdot 3^x} \right)dx\]
\[ \Rightarrow \int\left[ \left( \frac{2}{3} \right)^x + \left( \frac{3}{2} \right)^x + 2 \right]dx\]
\[ \Rightarrow \frac{\left( \frac{2}{3} \right)^x}{\text{ ln }\left( \frac{2}{3} \right)} + \frac{\left( \frac{3}{2} \right)^x}{\text{ln } \frac{3}{2}} + 2x + C ...........\left( \because \int a^x dx = \frac{a^x}{\text{ ln } a} \right)\]