Mathematics

# Evaluate the definite integral :$\displaystyle \int_{1}^{2}\dfrac {1}{\sqrt {(x-1) (2-x)}}dx$

##### SOLUTION
$I=\displaystyle \int_{1}^{2} \dfrac {1}{\sqrt {-x^2 +3x-2}}dx =\displaystyle \int_{1}^{2} \dfrac {1}{\sqrt {-\left\{ \left(x-\dfrac {3}{2}\right)^2-\left(\dfrac {1}{2}\right)^2 \right\}}}dx =\displaystyle \int_{1}^{2} \dfrac {1}{\sqrt {\left (\dfrac {1}{2}\right)^2-\left (x-\dfrac {3}{2}\right)^2}}dx$

Using,   $\displaystyle \int \dfrac 1{\sqrt {a^2-x^2}}=\sin ^{-1}\dfrac xa$

Replacing, we get

$\displaystyle \int _1^2\dfrac 1{\sqrt {\left(\dfrac 12\right)^2-\left(x-\dfrac 32\right)^2}}$

$=\left| \sin ^{-1}\left(\dfrac {x-\dfrac {3}{2}}{\dfrac 12}\right)\right|_1^2$

$\left| \sin^{-1} (2x-3)\right|_1^2=\sin ^{-1}(1)-\sin ^{-1}(-1)$

$=\dfrac {\pi}2+\dfrac {\pi}2$

$=\pi$

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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
The value of $\displaystyle \int_{2}^{3} \dfrac {\sqrt {x}}{\sqrt {5 - x} + \sqrt {x}}dx$ is
• A. $1$
• B. $2$
• C. None of these
• D. $\dfrac {1}{2}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle\int^{1}_{-1}x^3(1-x^2)dx=?$
• A. $-\dfrac{40}{3}$
• B. $\dfrac{40}{3}$
• C. $\dfrac{5}{6}$
• D. $0$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\int_0^{ \pi /2 } \sin^5 x \cos ^6 x dx =$
• A. $\dfrac {32}{693}$
• B. $\dfrac {8}{99}$
• C. $\dfrac {16}{63}$
• D. $\dfrac {8}{693}$

1 Verified Answer | Published on 17th 09, 2020

Q4 One Word Hard
If $\displaystyle I=\int \frac{dx}{2x\sqrt{1-x}\sqrt{2-x+\sqrt{1-x}}}$
$=\displaystyle -\dfrac{1}{2\sqrt{3}}\log \left | u+\dfrac{1}{2}+\sqrt{u^{2}+u+\frac{1}{3}} \right |+\frac{K}{16}\log \left | v-\frac{1}{2}+\sqrt{v^{2}-v+1} \right |+C$
where $\displaystyle u=\frac{1}{\sqrt{1-x}-1}$ and $v=\dfrac{1}{\sqrt{1-x}+1}$ then K is equal to

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
The value of $\displaystyle \int_0^{\dfrac{\pi}{2}}\dfrac{\sqrt{\cot t}}{\sqrt{\cot t}+\sqrt{\tan t}}dt$
• A. $\dfrac{\pi}{2}$
• B. $\dfrac{\pi}{6}$
• C. $\dfrac{\pi}{8}$
• D. $\dfrac{\pi}{4}$