Write a value


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


Let, $3+a^{x}=t$

Differentiating both sides with respect to $x$

$\frac{d t}{d x}=a^{x} \log a$

$\Rightarrow \frac{d t}{\log a}=a^{x} d x$

$y=\int \frac{1}{(\log a) t} d t$

Use formula $\int \frac{1}{t} d t=\log t$

$y=\frac{\log t}{\log a}+c$

Again, put $t=3+a^{x}$

$y=\frac{\log \left(3+a^{x}\right)}{\log a}+c$

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