Mathematics

# $\displaystyle\int \dfrac{2x}{2+x-x^2}dx$.

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

$\displaystyle\int\dfrac{2x-1}{2+x-x^2}dx+\int\dfrac{dx}{2+x-x^2}$

$\displaystyle=\int\dfrac{-dz}{z}-\int\dfrac{dx}{x^2-x-2}$

$\displaystyle2-\log|z|-\int\dfrac{dx}{x^2-2;\dfrac{x}{2}+(\dfrac{1}{2})^2-2-(\dfrac{1}{2})^2}$

$\displaystyle=-\log|z+x-x^2|-\int\dfrac{dx}{{x-\dfrac{1}{2}}^2-{\dfrac{\sqrt{5}}{2}}^2}$

$=-\log|z+x-x^2|-\dfrac{\sqrt{2}}{2.\sqrt{5}}\log \left|\dfrac{x-\dfrac{1}{2}-\dfrac{\sqrt{5}}{2}}{x-\dfrac{1}{2}+\dfrac{\sqrt{5}}{2}}\right|+c$

$=-\log |z+x-x^2|-\dfrac{1}{\sqrt{10}}\log\left|\dfrac{x-1-\sqrt{5}}{x-1+\sqrt{5}}\right|+c$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Single Correct Medium
Let $F(x)$ be the primitive of $\displaystyle \frac{3x+2}{\sqrt{x-9}}$ w.r.t $x$ . If $F(10)=60$ then the value of $F(13)$, is
• A. $66$
• B. $248$
• C. $264$
• D. $132$

1 Verified Answer | Published on 17th 09, 2020

Q2 TRUE/FALSE Hard
$\displaystyle \int_0^{\pi} sin x =\displaystyle \lim_{n \rightarrow \infty} \displaystyle \Sigma_{i=1}^n sin \left(\dfrac{\pi i}{n}\right) \dfrac{\pi}{n}$
State whether the above statement is True or False?
• A. False
• B. True

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
Let $\displaystyle f\left ( x \right )= \frac{x}{(1+x^{n})}^{1/n}$ for $\displaystyle n \geq 2$ and $\displaystyle g\left ( x \right )= \underset{f occurs n times}{\underbrace{\left ( f\circ f\circ ...\circ f \right )\left ( x \right )}}.$ Then $\displaystyle \int x^{n-2}g\left ( x \right )dx$
• A. $\displaystyle \frac{1}{ n-1 }\left ( 1+nx^{n} \right )^{1-\frac{1}{n}}+k$
• B. $\displaystyle \frac{1}{n\left ( n+1 \right )}\left ( 1+nx^{n} \right )^{1+\frac{1}{n}}+k$
• C. $\displaystyle \frac{1}{ n+1}\left ( 1+nx^{n} \right )^{1+\frac{1}{n}}+k$
• D. $\displaystyle \frac{1}{n\left ( n-1 \right )}\left ( 1+nx^{n} \right )^{1-\frac{1}{n}}+k$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
$\displaystyle\int \:e^x\left(x^2-x+1\right)dx$

1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Hard
If $n\rightarrow \infty$ then the limit of series in $n$ can be evaluated by following the rule : $\displaystyle \lim_{n\rightarrow \infty}\sum_{r=an+b}^{cn+d}\frac{1}{n}f\left ( \frac{r}{n} \right )=\int_{a}^{c}f(x)dx,$
where in $LHS$, $\dfrac{r}{n}$ is replaced by $x$,
$\dfrac{1}{n}$ by $dx$
and the lower and upper limits are $\lim_{n\rightarrow \infty }\dfrac{an+b}{n}\, and \, \lim_{n\rightarrow \infty }\dfrac{cn+d}{n}$ respectively.