Question

Write a value of $\int \frac{a^x}{3+a^x}dx$.

Answer 1

$$\begin{gathered} \mathrm{Let}I=\int \frac{a^{x}dx}{3+a^x} \\ \mathrm{Let}3+a^x=t \\ \Rightarrow a^x.\log adx=dt \\ \Rightarrow a^{x}dx=\frac{dt}{\log a} \\ \therefore I=\frac{1}{\log a}\int \frac{dt}{t} \\ =\frac{1}{\log a}\log t+C \\ =\frac{1}{\log a}\log \left(3+a^x\right)+\mathit{\text{C}}\left(\mathit{\text{∵ t}}\mathit{\text{=}}\mathit{3}\mathit{+}{\mathit{a}}^{\mathit{x}}\right) \end{gathered}$$