## Problem Statement

Calculate the definite integral:

$$
\int_{0}^{5} e^{\sqrt{(5-x) /(5+x)}} \cdot \frac{d x}{(5+x) \sqrt{25-x^{2}}}
$$
## Solution

$$
\int_{0}^{5} e^{\sqrt{(5-x) /(5+x)}} \cdot \frac{d x}{(5+x) \sqrt{25-x^{2}}}=
$$

Substitution:

$$
\begin{aligned}
& t=\sqrt{\frac{5-x}{5+x}} \\
& d t=\frac{1}{2} \cdot \sqrt{\frac{5+x}{5-x}} \cdot\left(\frac{5-x}{5+x}\right)^{\prime} \cdot d x=\frac{1}{2} \cdot \sqrt{\frac{5+x}{5-x}} \cdot \frac{-10}{(5+x)^{2}} \cdot d x=-\frac{5 d x}{\sqrt{25-x^{2}}(5+x)} \\
& x=0 \Rightarrow t=\sqrt{\frac{5-0}{5+0}}=1 \\
& x=5 \Rightarrow t=\sqrt{\frac{5-5}{5+5}}=0
\end{aligned}
$$

We get:

$$
=\int_{1}^{0} e^{t} \cdot\left(\frac{-d t}{5}\right)=-\frac{1}{5} \cdot \int_{1}^{0} e^{t} d t=-\left.\frac{e^{t}}{5}\right|_{1} ^{0}=-\frac{e^{0}}{5}+\frac{e^{1}}{5}=\frac{e-1}{5}
$$

Source — "http://pluspi.org/wiki/index.php/%D0%9A%D1%83%D0%B7%D0%BD%D0%B5%D1%86%D0%BE%D0%B2_%D0%98%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB%D1%8B_11-5"

Categories: Kuznetsov's Problem Book Integrals Problem 11 | Integrals

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