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 $$x = \cos 2\theta $$

$$dx =  - 2\sin 2\theta d\theta $$

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

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

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

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

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

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

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

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

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

$$=\pi $$

View Full Answer

Its FREE, you're just one step away


Subjective Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84
Enroll Now For FREE

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. $$ \displaystyle \frac{x}{\log x}+c$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
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. $$\pi$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
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. $$\dfrac{5}{9}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Easy
Evaluate:
$$ \int_{}^{} {\frac{{ - 1}}{{\sqrt {1 - {x^2}} }}dx} $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer