Mathematics

# $\displaystyle \int { \dfrac { \left( x+2 \right) dx }{ \sqrt { \left( x-2 \right) \left( x-3 \right) } } }$ is equal to:

$\sqrt{x^2-5x+6}+\frac{9}{2}\log\left(x-\frac{5}{2}\right)+\sqrt{x^2-5x+6}$

##### SOLUTION
$\int\frac{x+2.dx}{\sqrt{(x-2)(x-3)}}$

$=\int\frac{x+2}{\sqrt{x^2-5x+6}}.dx$

$\because x+2=p.(\frac{d}{dx}.x^2-5x+6)+q$

$=(2x-5)p+q$

$=2px-(5p-q)$

$=2p=1 , 5p-q=-2.$

$=p=\frac{1}{2}, \frac{5}{2}-q=-2$

$=q=\frac{9}{2}$

$\int\frac{\frac{1}{2}(2x-5)}{\sqrt{x^2-5x+6}}dx+\int\frac{\frac{9}{2}}{\sqrt{x^2-5x+6}}.dx$

$=\frac{1}{2}\int\frac{2x-5}{\sqrt{x^2-5x+6}}.dx+\frac{9}{2}\int\frac{1}{x^2-2.\frac{5}{2}x+\frac{25}{4}-\frac{24}{4+6}}.dx$

$=\frac{1}{2}\int\frac{2x-5}{\sqrt{x^2-5x+6}}.dx+\frac{9}{2}\int\frac{1}{\sqrt{(x-\frac{5}{2})^2-(\frac{1}{2}})^2}$

Let
$x^2-5x+6=t$

$=(2x-5)dx=dt$

$=\frac{1}{2}\int \frac{dt}{\sqrt{t}}+\frac{9}{2}\int\frac{1}{\sqrt{{(x-\frac{5}{2})^2}-(\frac{1}{2})^2}}$

$=\frac{1}{2}.\frac{t^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}+\frac{9}{2}.log[(x-\frac{5}{2})+\sqrt{x^2-5x+6}]$

$=\sqrt{x^2-5x+6}+\frac{9}{2}.log(x-\frac{5}{2})+\sqrt{x^2-5x+6}$

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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
$\displaystyle \frac{x^{2}-x-1}{x^{3}-8}=\displaystyle \frac{A}{x-2}+\displaystyle \frac{Bx+C}{x^{2}+2x+4}\Rightarrow A+B=$
• A. $0$
• B. $-1$
• C. $2$
• D. $1$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int \dfrac1ne^{x/n}dx$ is
• A. $e$
• B. $e^x$
• C. $ne^x$
• D. $e^{x/n}$

1 Verified Answer | Published on 17th 09, 2020

Q3 One Word Medium
$\displaystyle\int x^{3}\left ( \log x \right )^{2}dx=\frac{x^{4}}{4}\left ( \log x \right )^{2}-\frac{1}{8}x^{4}\log x+\frac{1}{4k}x^{4}.$ Find the value of $k$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
$\displaystyle\int^{\pi}_0\sqrt{|\cos x|-|\cos^3x|}dx$ is?
• A. $1$
• B. $3$
• C. $4$
• D. $\dfrac{4}{3}$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
$\displaystyle \int_2^3\dfrac{dx}{x^2-1}$
• A. $log\dfrac{3}{2}$
• B. $2log\dfrac{3}{2}$
• C. $\log\dfrac{3}{2}$
• D. $\dfrac{1}{2}log\dfrac{3}{2}$