Evalute : e^x e^ 2 x +4 e^x+13 d x

Question

Evalute : $\int \frac{e^x}{\sqrt{e^{2 x}+4 e^x+13}} d x$

✓

Answer

Let $I=\int \frac{e^x}{\sqrt{e^{2 x}+4 e^x+13}} d x$
Put $e^x d x \quad \therefore e^x d x=d t$
$
\begin{aligned}
\therefore I & =\int \frac{1}{\sqrt{t^2+4 t+13}} d t \\
& =\int \frac{1}{\sqrt{\left(t^2+4 t+4\right)+9}} d t \\
& =\int \frac{1}{\sqrt{(t+2)^2+(3)^2}} d t \\
& =\log \left|(t+2)+\sqrt{(t+2)^2+(3)^2}\right|+c \\
& =\log \mid(t+2)+\sqrt{t^2+4 t+13 \mid}+c \\
& =\log \left|\left(e^x+2\right)+\sqrt{e^{2 x}+4 e^x+13}\right|+c .
\end{aligned}
$

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