Mathematics

# Integrate $\displaystyle \int \sqrt{\dfrac{a - x}{a + x}} dx$

##### SOLUTION
$\int \sqrt{\dfrac{a-x}{a+x}}dx$

substitute $x=a\cos \theta \rightarrow dx=-a\sin \theta d\theta$

$-\int \sqrt{\dfrac{a-a\cos \theta}{a+a\cos \theta}}-a\sin \theta d\theta$

$-\int \sqrt{\dfrac{a(2\sin ^2\frac{\theta }{2})}{a(2\cos ^2\frac{\theta }{2})}}-a2\sin \dfrac{\theta }{2}\cos \dfrac{\theta }{2} d\theta$

$-2a\int \sin ^2\dfrac{\theta }{2}d\theta$

$-2a\int \dfrac{1}{2}(1-\cos \theta )d\theta$

$-a[\theta -\sin \theta ]+C$

$-a\left [ \cos^{-1}\left ( \dfrac{x}{a} \right )-\sqrt{1-\left ( \dfrac{x}{a} \right )^2} \right ]+C$

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

#### Realted Questions

Q1 Subjective Medium
Evaluate the following integral:
$\displaystyle \int { { x }^{ 3 }\cos { \left( { x }^{ 4 }+1 \right) } } dx\quad$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
If $\displaystyle \int_{0}^{1} \frac{\tan^{-1} x}{x}dx$  is equal to
• A. $\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{\sin x}{x}dx$
• B. $\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{x}{\sin x}dx$
• C. $\displaystyle \frac{1}{2} \int_{0}^{\frac{\pi}{2}}\frac{\sin x}{x}dx$
• D. $\displaystyle \frac{1}{2} \int_{0}^{\frac{\pi}{2}}\frac{x}{\sin x}dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
$\int { \cfrac { 2{ x }^{ 3 } }{ \left( 4+{ x }^{ 8 } \right) } } dx=$?
• A. $\cfrac { 1 }{ 2 } \tan ^{ -1 }{ \cfrac { { x }^{ 4 } }{ 2 } } +C$
• B. $\cfrac { 1 }{ 2 } \tan ^{ -1 }{ { x }^{ 4 } } +C\quad$
• C. none of these
• D. $\cfrac { 1 }{ 4 } \tan ^{ -1 }{ \cfrac { { x }^{ 4 } }{ 2 } } +C\quad$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\displaystyle \int\frac{x^{1/3}}{(1+x^{4/3})^{2}}dx=$
• A. $\displaystyle \frac{3}{4(1+x^{4/3})}+c$
• B. $\displaystyle \frac{1}{(1+x^{4/3})}+c$
• C. $\displaystyle \frac{-1}{(1+x^{4/3})}+c$
• D. $\displaystyle \frac{-3}{4(1+x^{4/3})}+c$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
$\displaystyle \int \dfrac{dx}{x^2 + 2x + 2} = f(x) + c \Longrightarrow f(x) =$
• A. $2\tan^{-1}(x + 1)$
• B. $-\tan^{-1}(x + 1)$
• C. $3\tan^{-1}(x + 1)$
• D. $\tan^{-1}(x + 1)$