Mathematics

# If $\int {\sqrt {\dfrac{{1 - x}}{{1 + x}}} dx = \sqrt {1 - {x^2}} } + f\left( x \right) + c,\,x \in [0,\,1)$ where $f\left( 0 \right) = - \dfrac{\pi }{2}$ then $f\left( {\dfrac{1}{2}} \right)$ is ______

$- \dfrac{\pi }{3}$

##### SOLUTION
Solution -

$=\displaystyle \int \sqrt{\dfrac{1-x}{1+x}}\times \dfrac{\sqrt{1-x}}{\sqrt{1-x}}dx$

$= \displaystyle\int \dfrac{1-x}{\sqrt{1-x^{2}}}dx = \int \dfrac{dx}{\sqrt{1-x^{2}}}-\int \dfrac{x\,dx}{\sqrt{1-x^{2}}}$

$=\displaystyle -cos^{-1}x-\int \dfrac{xdx}{\sqrt{1-x^{2}}}$

put $1-x^{2} = t$

$-2\,x\,dx = dt$

$=\displaystyle -cos^{-1}x+\dfrac{1}{2}\int \dfrac{dt}{\sqrt{t}}$

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

$= f(x) = -cos^{-1}x$

$f(0) = \dfrac{-\pi }{2}$   $f(\dfrac{1}{2} )= -cos^{-1}(\dfrac{1}{2}) = -\dfrac{\pi }{3}$

A is correct

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

#### Realted Questions

Q1 Single Correct Medium
Find  $\int \cfrac {dx} {{x}^{2}-{a}^{2}}$ are hence evaluate
• A. $\int \cfrac {dx} {{x}^{2}- 8x+5}$
• B. $\int \cfrac {dx} {{3x}^{2}+13x-10}$
• C. None of these
• D. $\int \cfrac {dx} {{x}^{2}-36}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
If $\displaystyle \int \log \left ( x^{2}+x \right )dx=x\log x+\left ( x+1 \right )\log \left ( x+1 \right )+K,$ then K is equal to

• A. $\displaystyle 2x+\log\left ( x+1 \right )+C$
• B. $\displaystyle 2x-\log\left ( x+1 \right )+C$
• C. constant
• D. none of these

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
$\displaystyle \int_{e^{e^{e}}}^{e^{e^{e^{e}}}}\frac{dx}{xlnx\cdot ln\left ( lnx \right )\cdot ln\left ( ln\left ( lnx \right ) \right )}$ equals
• A. 1/e
• B. e-1
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1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
If $\phi \left( x \right) = {\int {\left\{ {\phi \left( x \right)} \right\}} ^{ - 2}}dx$ and $\phi \left( 1 \right) = 0$ then $\phi \left( x \right) =$
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Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$