Mathematics

# Evaluate $\displaystyle \int_{0}^{2}\dfrac {1}{4+x-x^2}dx$

##### SOLUTION

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

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

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

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

Using $\displaystyle \int \dfrac{1}{{a^2-x^2}}dx =\dfrac{1}{2a}\log|\dfrac{{a+x}}{a-x} |+c$

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

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

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

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

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

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

#### Realted Questions

Q1 Subjective Medium
$\int {\left( {3x - 2} \right)\sqrt {{x^2} + x + 1} dx} =$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Integrate the function    $\sqrt {x^2+4x+1}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
If $\displaystyle \int \frac{2dx}{\left [ \left ( x-5 \right ) +\left ( x-7 \right )\right ]\sqrt{\left ( x-5 \right )\left ( x-7 \right )}}=f\left [ g\left ( x \right ) \right ]+c,$ then

• A. $\displaystyle f\left ( x \right )=\sin ^{-1}x,g\left ( x \right )=\sqrt{\left ( x-5 \right )\left ( x-7 \right )}$
• B. $\displaystyle f\left ( x \right )=\sin ^{-1}x,g\left ( x \right )=\left ( x-5 \right )\left ( x-7 \right )$
• C. $\displaystyle f\left ( x \right )=\tan ^{-1}x,g\left ( x \right )=\left ( x-5 \right )\left ( x-7 \right )$
• D. $\displaystyle f\left ( x \right )=\tan ^{-1}x,g\left ( x \right )=\sqrt{\left ( x-5 \right )\left ( x-7 \right )}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium

$\displaystyle \int_{0}^{1}\frac{dx}{x+\sqrt{x}}=$
• A. log 2
• B. 3 log 3
• C. $\displaystyle \frac{1}{2}$ log2
• D. 2 log 2

Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$