Question

Evaluate: $\int \frac{dx}{2^x+3}$

Answer 1

$I=\int \frac{dx}{2^x+3}$

$I=\frac{1}{3}\int \frac{(2^x+3-2^x)dx}{2^x+3}$

$I=\frac{1}{3}\int dx-\frac{1}{3}\int \frac{2^{x}dx}{2^x+3}$

Let $t=2^x+3\Rightarrow dt=2^x\log 2dx$

$\Rightarrow \frac{dt}{\log 2}=2^{x}dx$

$=\frac{1}{3}\int dx-\frac{1}{3}\int \frac{2^{x}dx}{2^x+3}$

$=\frac{x}{3}-\frac{1}{3\log 2}\int \frac{dt}{t}$

$=\frac{x}{3}-\frac{1}{3\log 2}\log t+c$ where $t=2^x$

$=\frac{x}{3}-\frac{1}{3\log 2}\log 2^x+c$

$=\frac{x}{3}-\frac{x\log 2}{2\log 2}+c$

$=\frac{x}{3}-\frac{x}{2}+c$

$=\frac{2x-3x}{6}+c$

$=\frac{-x}{6}+c$