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