## Problem Statement

Calculate the definite integral:

$$
\int_{0}^{1} e^{\sqrt{(1-x) /(1+x)}} \cdot \frac{d x}{(1+x) \sqrt{1-x^{2}}}
$$
## Solution

$$
\int_{0}^{1} e^{\sqrt{(1-x) /(1+x)}} \cdot \frac{d x}{(1+x) \sqrt{1-x^{2}}}=
$$

Substitution:

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

We get:

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

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_11-7»

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

- Last edited on this page: 15:11, 17 June 2009.
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## Problem Kuznetsov Integrals 11-8

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