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$Evaluation of $\int_{-\infty}^{\infty}\frac{1}{(x^2+ax+a^2)(x^2+bx+b^2)}dx$ Although we have solve it using partial function Decomposition, But can we solve without using partial Decomposition. If yes then How can we solve it, Thanks

$If $f$ is a double differentiable function and satisfy the condition$ f(0)=0$ and $f(1)=0$ and $\frac{d^2}{dx^2}\left(e^{-x}f(x)-x^2\right)>0\;\forall x\in (0,1)$ $Then number of values of $x$ for which $f(x)-3=(x^2-x)e^x\;\forall x\in (0,1)$...

Thanks Paradoxica for Nice explanation.

To Paradoxica I did not understand it. Would you like to explain me in detail, Thanks

Thanks InteGrand. But when we solve $PA+PB=1\;,$ Then I am getting Hyperbola. Where i have done wrong in my calculation

$If $P(x,y)$ be a variable point in $x-y$ plane such that $PA+PB=1\;,$ $Where $A(-1,0)$ and $B(1,0)\;,$Then locus of point $P$ is ................................................................................................................ $Given $PA+PB=1\;,$ Then $(PA)^2-(PB)^2=PA-PB$...

Re: MX2 2016 Integration Marathon $Let $I = \int\frac{1}{(x^4-1)^2}dx = \frac{1}{4}\int \left[\frac{1}{x^3}\cdot \frac{4x^3}{(x^4-1)^2}\right]dx$ $Using Integration by parts, We get$ $So $I = -\frac{1}{4}\cdot \frac{1}{x^4-1}-\frac{3}{4}\int\frac{1}{x^4\cdot (x^4-1)}dx$ $So $I =...

Re: MX2 2016 Integration Marathon $Evaluation of $\int_{0}^{1}\left(tx+1-x\right)^ndx\;,$ Where $n\in \mathbb{Z^{+}}$ $ and $t$ is a parameter independent of $x$ $Hence show that $\int_{0}^{1}x^k\left(1-x\right)^{n-k}dx = \left[\binom{n}{k}(n+1)\right]^{-1}$ for $k=0,1,2,3,.....,n$

Re: MX2 2016 Integration Marathon $If $a,b>0\;,$ Then $\int_{0}^{\pi}\frac{\sin^2 x}{a^2-2ab\cos x+b^2}dx\;,$ $and value of above integral if $b>a>0$

Re: MX2 2016 Integration Marathon $Let $I = \int\frac{x^4+1+x^2-x^2}{x^6-1} = \int\frac{x^4+1+x^2}{x^6-1}dx-\int\frac{x^2}{x^6-1}dx$ $Let $I = \int\frac{1}{x^2-1}dx-\frac{1}{3}\int\frac{(x^3)^{'}}{(x^3)^2-1}dx$ $So $I = \frac{1}{2}\ln\left|\frac{x-1}{x+1}\right|-\frac{1}{3}\cdot...

Re: HSC 2016 4U Marathon $Numbers of ordered pair,s $(z,\omega)$ of the complex number $z$ and $\omega$ satisfying$ $the system of equations $z^3+\bar{\omega}^7=0$ and $z^5\cdot \omega^{11} = 1$

$If $f(x)$ is a twice differentiable function such that $f(x)+f''(x)=-x|\sin x|f'(x)$ $Where $x\geq 0$ and $f(0) = -3$ and $f'(0)=4\;,$ $Then maximum value of $f(x)$

Re: MX2 2016 Integration Marathon $Let $\alpha \in \mathbb{R^{+}}$ and $f(\alpha) = \int_{0}^{\infty}\frac{\ln x}{x^2+\alpha x+\alpha^2}dx$ $and $\alpha f(\alpha)-f(1) = \frac{\pi}{\sqrt{3}}\;,$ Then $\alpha $

Re: MX2 2016 Integration Marathon $Evaluation of $\int\frac{1}{(x^4-1)^2}dx$ and $\int_{0}^{1}\frac{x\ln(x)}{\sqrt{1-x^2}}dx$

Re: MX2 2016 Integration Marathon $Put $x-a+x-b = 2t\Rightarrow t=x-\frac{a+b}{2}\;,$ Then $dt=dx$ $So $I = \int\frac{\sin \left(t+A\right)}{\sin(t+B)\cdot \sin(t-B)}dx\;,$ Where $\frac{a+b}{2}=A$ and $\frac{a-b}{2}=B$ $So $I = \int\frac{\sin t\cdot \cos A+\cos t\cdot \sin...

Re: MX2 2016 Integration Marathon $If $I = \int_{0}^{1}(1-x^{50})^{100}dx$ and $J = \int_{0}^{1}(1-x^{50})^{101}dx\;,$ Then $\frac{I}{J} = $

Re: MX2 2016 Integration Marathon $Put $x=\frac{a^2+b^2}{y}\;,$ Then $dx = -\frac{a^2+b^2}{y^2}$ and changing Limits,We get I = \int_{\infty}^{0}\frac{\ln\left(\frac{a^2+b^2}{y}\right)}{\left(\frac{a^2+b^2}{y}+a\right)^2+b^2}\times -\frac{(a^2+b^2)}{y^2}dy $So $I =...

Re: MX2 2016 Integration Marathon (1)\; $Evaluate $I = \int_{0}^{1}4x^3\left\{\frac{d^2}{dx^2}(1-x^2)^5\right\}dx$ (2)\;$If $I = \int_{0}^{4\pi}e^t\left(\sin^6 t+\cos^4 t\right)dt$ and $J = \int_{0}^{\pi}e^t\left(\sin^6 t+\cos^4 t\right)dt\;,$ $Then $\frac{I}{J}\;,$Where $a\in \mathbb{N}$

Re: MX2 2016 Integration Marathon $Let $I = \int\frac{\sin x+\cos x}{\sqrt{\sin x\cos x}}dx = \sqrt{2}\int\frac{\sin x+\cos x}{\sqrt{\sin 2x}}dx$ $Now put $\sin x-\cos x = t\Rightarrow 1-t^2=\sin 2x\;,$ Then $(\cos x+\sin x)dx = dt$ $So Integral $I =...

$Number of permutations of $1,2,3,4,5,6,,7,8,9$ such that $1$ appear$ $somewhere to the left of $2$ and $3$ appear to the left of $4$ and $5$ $somewhere to the left of $6,$ are$ $like $815723946$ would be one such permutation$