Mathematics

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

##### SOLUTION
Consider, $I=\displaystyle \int_0^2 \dfrac {1}{4+x-x^2}dx$

$=-\displaystyle \int_0^2 \dfrac {1}{x^2 -x-4}dx$

$=-\displaystyle \int_0^2\dfrac {1}{x^2-x-\dfrac {1}{4}-\dfrac {{17}}{4}}dx$

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

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

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

$\Rightarrow \ I=\dfrac {1}{\sqrt {17}}\left\{\dfrac {\sqrt {17}+3}{\sqrt {17}-3} -\log \dfrac {\sqrt {17}-1}{\sqrt {17}+1} \right\}$

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

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

$=\dfrac {1}{\sqrt {17}}\log \left (\dfrac {21+5\sqrt {17}}{4}\right)$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
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#### Realted Questions

Q1 Subjective Hard
Evaluate $\int_{0}^{4}(|x| + |x - 2| + |x - 4|)$dx.

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
Value of the integral $\displaystyle \int_{0}^{\pi}\frac{xdx}{1+\sin x}$, is
• A. $\displaystyle \frac{\pi}{6}$
• B. $\displaystyle \frac{\pi}{15}$
• C. $4\pi$
• D. $\pi$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
The value of $\displaystyle\int _{ 0 }^{ \infty }{ \frac { \log { x } }{ { a }^{ 2 }+{ x }^{ 2 } } dx }$
• A. $\dfrac{\pi \log a}{a}$
• B. $\pi \log a$
• C. $0$
• D. $\dfrac{2\pi \log a}{a}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
$\int \dfrac{dx}{x(x^5+3)}$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
The function$int {:R \to \left[ { - \frac{1}{2},\frac{1}{2}} \right]}$ defined as
$\int {(x) = \frac{x}{{1 + x2}}}$ is ( 2017 main offline)
• A. surjective but not injective
• B. neither injective nor surjective
• C. invertible
• D. Injective but not Surjective