Mathematics

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

$A+B=0$

##### SOLUTION
$\int{\dfrac{dx}{(x+1)(x-2)}}$

This can be written as

$\Rightarrow \int{(\dfrac{-1}{3}\dfrac{1}{x+1}+\dfrac{1}{3}\dfrac{1}{x-2})}dx$

$\Rightarrow \dfrac{-1}{3}\int {\dfrac{1}{x+1}dx +\dfrac{1}{3}\int\dfrac{1}{x-2}}dx$

$\Rightarrow \dfrac{-1}{3}\log(x+1)+\dfrac{1}{3}\log(x-2)+C$

Thus comparing this whith given expression,

$\Rightarrow A=\dfrac{-1}{3}, B=\dfrac{1}{3}$

$\Rightarrow A+B=0$

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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
The antiderivative of $f(x)=\dfrac{1}{3+5\sin x+3\cos x}$ whose graph passes through the point $(0,0)$ is
• A. $\dfrac{1}{5}\left(\log \left|1-\dfrac{5}{3}\tan x/2\right|\right)$
• B. $\dfrac{1}{5}\left(\log \left|1+\dfrac{5}{3}\cot x/2\right|\right)$
• C. $None\ of\ these$
• D. $\dfrac{1}{5}\left(\log \left|1+\dfrac{5}{3}\tan x/2\right|\right)$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
If $\displaystyle \int _{ 0 }^{ \pi /2 }{ \dfrac { \sin { x } }{ 1+\sin { x+\cos { x } } } }dx=\dfrac{\pi}{a}-\dfrac{1}{2}lnb,\ a,b \in N$ then
• A. $a+b=4$
• B. $a-b=4$
• C. $a-b=6$
• D. $a+b=6$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Hard
Evaluate:
$\int { \cfrac { 2x }{ ({ x }^{ 2 }+1)({ x }^{ 2 }+3) } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 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. $\displaystyle \frac {1}{a - b} \sqrt {a \sin^2 x + b \cos^2 x} + C$

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.