Mathematics

# $\displaystyle \int \frac{dx}{\left ( x+1 \right )\left ( x-2 \right )}=A\log \left ( x+1 \right )+B\log \left ( x-2 \right )+C,$ where

$\displaystyle A+B=0$

$\displaystyle A : B=-1$

##### SOLUTION

$\displaystyle \int \frac { dx }{ \left( x+1 \right) \left( x-2 \right) }$

$\displaystyle =\frac { 1 }{ 3 } \int { \frac { \left( x+1 \right) -\left( x-2 \right) }{ \left( x+1 \right) \left( x-2 \right) } } dx$

$\displaystyle =\frac { 1 }{ 3 } \left[ \int { \frac { 1 }{ x-2 } dx } -\int { \frac { 1 }{ x+1 } dx } \right]$

$\displaystyle =\frac { 1 }{ 3 } \log { \left( x-2 \right) } -\frac { 1 }{ 3 } \log { \left( x+1 \right) } +C$

Hence, we get $A = -\dfrac13$ and $B = \dfrac13$

$\therefore A+B = 0$ and $A:B = -1$

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

#### Realted Questions

Q1 Single Correct Medium
$\displaystyle \int {\frac{{xdx}}{{\sqrt {1 + {x^2} + \sqrt {{{(1 + {x^2})}^3}} } }}}$ is equal to :
• A. $\frac{1}{2}\ln (1 + \sqrt {1 + {x^2}} ) + c$
• B. $2(1 + \sqrt {1 + {x^2}} ) + c$
• C. none of these
• D. $2\sqrt {1 + \sqrt {(1 + {x^2})} } + c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int _{ 0 }^{ a }{ \frac { dx }{ a+\sqrt { { a }^{ 2 }-{ x }^{ 2 } } } }$ is equal to
• A. $\displaystyle \frac { \pi }{ 2 } +1$
• B. $\displaystyle 1-\frac { \pi }{ 2 }$
• C. none of these
• D. $\displaystyle \frac { \pi }{ 2 } -1$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
If $\displaystyle I = \int \frac {\cos x \: dx}{\sqrt {a + b \cot^2 x}} (a > b > 0)$, then I equals
• A. $\displaystyle \frac {1}{a - b} \sqrt {a + b \cot^2 x} + C$
• B. $\displaystyle \frac {1}{a - b} \left ( \sqrt {a + b \cot^2 x} + x \right ) + C$
• C. $\displaystyle \frac {1}{a - b} \left ( \sqrt {a + b \cot^2 x} - x \right ) + C$
• D. $\displaystyle \frac {1}{a - b} \sqrt {a \sin^2 x + b \cos^2 x} + C$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
If $\displaystyle I_1= \int_{0}^{\frac{\pi}{2}}\frac{\cos^2x}{1+\cos^2x} \:dx,$

$\displaystyle I_2=\int_{0}^{\frac{\pi}{2 }}\frac{\sin^2 x}{1+\sin^2 x}\:dx,$

$\displaystyle I_3=\int_{0}^{\frac{\pi}{2}}\frac{1+2\cos^2x \sin^2x}{4+2\cos^2 x\sin^2 x}\: dx$, then
• A. $I_1=I_2>I_3$
• B. $I_3>I_1=I_2$
• C. None of these
• D. $I_1=I_2=I_3$

Let $n \space\epsilon \space N$ & the A.M., G.M., H.M. & the root mean square of $n$ numbers $2n+1, 2n+2, ...,$ up to $n^{th}$ number are $A_{n}$, $G_{n}$, $H_{n}$ and $R_{n}$ respectively.