Mathematics

# Evaluate the following definite integral :$\displaystyle \int_{0}^{2}\dfrac {1}{4+x-x^2}dx$

##### SOLUTION
$I=\displaystyle \int_0^2 \dfrac {1}{4+x-x^2}dx=-\displaystyle \int_0^2 \dfrac {1}{x^2 -x-4}dx =-\displaystyle \int_0^2\dfrac {1}{\left (x-\dfrac {1}{2}\right)^2-\left (\dfrac {\sqrt {17}}{2}\right)}dx$

$\Rightarrow \ I=\displaystyle \int_0^2 \dfrac {1}{\left (\dfrac {\sqrt {17}}{2}\right)-\left (x-\dfrac {1}{2}\right)^2}dx =\dfrac {1}{\sqrt {17}} \left [\log \left (\dfrac {\sqrt {17}+2x-1}{\sqrt {17} -2x+1}\right) \right]_0^2$

$\Rightarrow \ I=\dfrac {1}{\sqrt {17}}\left\{\dfrac {\sqrt {17}+3}{\sqrt {17}-3} -\log \dfrac {\sqrt {17}-1}{\sqrt {17}+1} \right\}=\dfrac {1}{\sqrt {17}}\log \left (\dfrac {52+12\sqrt {17}}{18-2\sqrt {17}} \times \dfrac {18+2\sqrt {17}}{18+2\sqrt {17}}\right)$

$\Rightarrow \ I=\dfrac {1}{\sqrt {17}}\log \left (\dfrac {1344+320\sqrt {17}}{256}\right)=\dfrac {1}{\sqrt {17}}\log \left (\dfrac {21+5\sqrt {17}}{4}\right)$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Medium
The function $\displaystyle F(x)=\int_{0}^{x}log\frac{(1-x)}{1+x}dx$ is a function that is
• A. Odd
• B. Periodic
• C. none of these
• D. Even

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Hard
Evaluate $\int\limits_0^{\frac{\pi }{4}} {{{\cos }^{\frac{3}{2}}}\left( {2x} \right)cos\left( x \right)dx}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
If $\int { \cfrac { \left( { x }^{ 2 }+1 \right) \left( { x }^{ 2 }+4 \right) }{ \left( { x }^{ 2 }+3 \right) \left( { x }^{ 2 }-5 \right) } } dx=\int { \left\{ 1+\cfrac { f(x) }{ \left( { x }^{ 2 }+3 \right) \left( { x }^{ 2 }-5 \right) } \right\} } dx$
$x+A\tan ^{ -1 }{ \left( \cfrac { x }{ A' } \right) } +B\log { \left( \cfrac { x-l }{ x+m } \right) } +K\quad$ then which of the following is correct
• A. $A=\cfrac { 1 }{ 4\sqrt { 3 } } ,B=\cfrac { 27 }{ 8\sqrt { 5 } } ,K\in R$
• B. $f(x)=7{ x }^{ 2 }+19,A'=\sqrt { 3 } ,K\in R$
• C. $l=m=\sqrt { 5 } ,L=1,K\in R$
• D. All of these

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
Number of positive continuous functions $f(x)$ defined in $[0,1]$ for which $\displaystyle\int_0^1{f(x)dx}=1$, $\displaystyle\int_0^1{xf(x)dx}=2$, $\displaystyle\int_0^1{x^2f(x)dx}=4$, is
• A. $1$
• B. $4$
• C. Infinite
• D. None of these

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
$4 \displaystyle \int \dfrac{\sqrt{a^6 + x^8}}{x} dx$ is equal to ______________________.
• A. $a^6 ln\vert \dfrac {\sqrt{a^6 + x^8} - a^3} {\sqrt {a^6 + x^8} + a^3}\vert + c$
• B. $\sqrt{a^6 + x^8} + \dfrac {a^3}{2} ln \vert \dfrac {\sqrt{a^6 + x^8} - a^3} {\sqrt {a^6 + x^8} + a^3}\vert + c$
• C. $a^6 ln \vert \dfrac {\sqrt{a^6 + x^8} + a^3} {\sqrt {a^6 + x^8} - a^3}\vert + c$
• D. $\sqrt{a^6 + x^8} + \dfrac{a^3}{2} ln \vert \dfrac {\sqrt{a^6 + x^8} + a^3} {\sqrt {a^6 + x^8} - a^3}\vert + c$