Question

Write a value of

\[\int\frac{a^x}{3 + a^x} \text{ dx}\]

Answer 1

\[\text{ Let I }= \int \frac{a^x dx}{3 + a^x}\]
\[\text{ Let 3} + a^x = t\]
\[ \Rightarrow a^x . \text{ log  a  dx }= dt\]
\[ \Rightarrow a^x dx = \frac{dt}{\log a}\]
\[ \therefore I = \frac{1}{\log a}\int\frac{dt}{t}\]
\[ = \frac{1}{\log a}\log t + C\]
\[ = \frac{1}{\log a}\log   \left( \text{ 3 }+ a^x \right) + C\left( \because t=3 + a^x \right)\]