Mathematics

# $\int \dfrac { x ^ { 2 } + x - 1 } { x ^ { 2 } + x - 6 } d x =$

$x - \log | x + 3 | + \log | x - 2 | + c$

##### SOLUTION
$\displaystyle \int \frac{x^{2}+x-1}{x^{2}+x-6}dx=?$

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

$\displaystyle =\int (1+\frac{5}{(x+3)(x-2)})dx$

$\displaystyle =\int 1dx+\frac{5}{5}\int [\frac{1}{x-2}-\frac{1}{x+3}]dx$

$=x+log(x-2)-log(x+3)+c$

$=x-log(x+3)+log(x-2)+c$

Option (b)

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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
$\displaystyle \int \frac{x^{2}\tan ^{-1}x}{1+x^{2}}dx$
• A. $\displaystyle \tan^{-1}x-\frac{1}{2}\log\left ( 1+x^{2} \right )-\frac{1}{2}\left ( \tan^{-1}x \right )^{2}.$
• B. $\displaystyle x\tan^{-1}x+\log\left ( 1+x^{2} \right )-\frac{1}{2}\left ( \tan^{-1}x \right )^{2}.$
• C. $\displaystyle x\tan^{-1}x-\frac{1}{2}\log\left ( 1+x^{2} \right )+\frac{1}{2}\left ( \tan^{-1}x \right )^{2}.$
• D. $\displaystyle x\tan^{-1}x-\frac{1}{2}\log\left ( 1+x^{2} \right )-\frac{1}{2}\left ( \tan^{-1}x \right )^{2}.$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
Let $[x]$ denote the largest integer not exceeding $x$ and $\left \{x\right \} = x - [x]$. Then
$\int_{0}^{2012} \dfrac {e^{\cos(\pi \left \{x\right \})}}{e^{\cos(\pi \left \{x\right \})} + e^{-\cos(\pi \left \{x\right \})}} dx$ is equal to.
• A. $0$
• B. $2012$
• C. $2012\pi$
• D. $1006$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
If $\int { \cfrac { 1 }{ \left( 1+x \right) \sqrt { x } } } dx=f\left( x \right) +A$, where A is any arbitary constant, then the function f(x) is
• A. $2\tan^{-1}{x}$
• B. $2\cot^{-1}{\sqrt{x}}$
• C. $log_{e}{(1+x)}$
• D. $2\tan^{-1}{\sqrt{x}}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
Evaluate the integral
$\displaystyle \int_{0}^{1}\tan^{-1}x \ dx$
• A. $\dfrac{\pi}{4}-\dfrac{1}{4} log2$
• B. $\displaystyle \frac{\pi}{4}+\frac{1}{2} log 2$
• C. $\displaystyle \frac{\pi}{4}+\frac{1}{4} log 2$
• D. $\displaystyle \frac{\pi}{4}-\frac{1}{2} log2$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
Value of the integral $\displaystyle \int { \cfrac { x^{ 2 }+5x-1 }{ \sqrt { x } } } dx$ is
• A. $\cfrac { 2 }{ 5 } { x }^{ 5/2 }+\cfrac { 2 }{ 3 } 5{ x }^{ 1/2 }-2{ x }^{ 1/2 }+C$
• B. $\cfrac { 3 }{ 5 } { x }^{ 5/2 }+\cfrac { 2 }{ 3 } 5{ x }^{ 1/2 }-2{ x }^{ 1/2 }+C$
• C. $\cfrac { 2 }{ 5 } { x }^{ 5/2 }+\cfrac { 2 }{ 3 } 5{ x }^{ 1/2 }-2{ x }^{ 1/2 }$
• D. $\cfrac { 2 }{ 5 } { x }^{ 5/2 }+\cfrac { 2 }{ 3 } 5{ x }^{ 3/2 }-2{ x }^{ 1/2 }+C$