Mathematics

# Integrate $\displaystyle\int {\sqrt {\frac{{1 + x}}{{1 - x}}} dx\ , on ( - 1,1)\,.}$

##### SOLUTION

$\displaystyle\int {\sqrt {{{1 + x} \over {1 - x}}} } dx$

Put $x = \cos 2\theta$

$dx = - 2\sin 2\theta d\theta$

$= - \displaystyle\int {\sqrt {{{1 + {{\cos }2}\theta } \over {1 - {{\cos }2}\theta }}} } \times 2\sin 2\theta d\theta$

$= - 2\int {\sqrt {{{\displaystyle2{{\cos }^2}\theta } \over {\displaystyle2{{\sin }^2}\theta }}} \times \sin 2\theta d\theta }$

$= - 2\displaystyle\int {{{\cos \theta } \over {\sin \theta }}} \times 2\sin \theta \cos \theta d\theta$

$= - 4\displaystyle\int {{{\cos }^2}\theta d\theta }$

$= - 2\int {\left( {1 + \cos 2\theta } \right)} d\theta = -2\theta - {{\displaystyle2\sin 2\theta } \over \displaystyle2}$

$= - {\cos ^{ - 1}}x - \sqrt {1 - {x^2}}$

$= - {\left[ {{{\cos }^{ - 1}}x + \sqrt {1 - {x^2}} } \right]_{ - 1}}^{_1}$

$=\left[ {\left( {{{\cos }^{ - 1}}1 + \pi } \right) - \left( {{{\cos }^{ - 1}} - 1 + 0} \right)} \right]$

$=\left[ {0 - \pi } \right]$

$=\pi$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Single Correct Medium
Evaluate $\displaystyle \int\frac{\log(x/e)}{(\log x)^{2}}dx$
• A. $\displaystyle \frac{\log x}{x}+c$
• B. $^{\dfrac{x}{log(x)^{2}}+c}$
• C. $\displaystyle \frac{(\log x)^{2}}{x}+c$
• D. $\displaystyle \frac{x}{\log x}+c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
Consider $I=\displaystyle \int^{\pi}_{0}\displaystyle\frac{xdx}{1+\sin x}$. What is I equal to?
• A. $-\pi$
• B. $0$
• C. $2\pi$
• D. $\pi$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
The $p.m.f.$ of a $r.v.X$ is
 $X=x$ $1$ $2$ $3$ $P(X=x)$ $1/5$ $2/5$ $2/5$
,then $E(X)=$
• A. $\dfrac{5}{7}$
• B. $\dfrac{11}{5}$
• C. $\dfrac{7}{5}$
• D. $\dfrac{5}{9}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate
$\displaystyle \int \dfrac {x^{3}}{\sqrt {1+2x^4}}dx$

$\int_{}^{} {\frac{{ - 1}}{{\sqrt {1 - {x^2}} }}dx}$