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Re: MX2 2016 Integration Marathon $Evaluation of $\int_{-\infty}^{\infty}\frac{1}{(x^2+ax+a^2)(x^2+bx+b^2)}dx\;,$ Given $a,b>0$

Re: MX2 2016 Integration Marathon I = \int\frac{(\sin^3 \theta-\cos^3 \theta-\cos^2 \theta)(\sin \theta+\cos \theta+\cos^2 \theta)^{2010}}{(\sin \theta \cdot \cos \theta)^{2012}}d\theta I = \int \left(\tan \theta \cdot \sec \theta -\cot \theta \cdot \csc \theta -\csc^2 \theta\right)\cdot (\sec...

Thanks Paradoxica would you like to explain me in detail.

$Sum of $n$ terms of series $\frac{1}{2}+\frac{1}{2!}\left(\frac{1}{2}\right)^2+\frac{1\cdot 3}{3!}\left(\frac{1}{2}\right)^3+\frac{1\cdot 3 \cdot 5}{4!}\left(\frac{1}{2}\right)^4+...$

Re: MX2 2016 Integration Marathon $Evaluation of $\int_{0}^{1}\frac{\sin \theta \left(\cos^2 -\cos^2 \frac{\pi}{5}\right)\cdot \left(\cos^2 -\cos^2 \frac{2\pi}{5}\right)}{\sin 5\theta}d\theta$

Re: MX2 2016 Integration Marathon $Let $I = \int\frac{2\cos \theta}{25+16\sin 2\theta}d\theta = \int\frac{\cos \theta+\sin\theta}{25+16\sin 2\theta}d\theta + \int\frac{\cos \theta-\sin \theta}{25+16\sin 2\theta}d\theta$ $So $I = \int\frac{\cos \theta+\sin \theta}{25+16\left[1-(\sin \theta-\cos...

Re: MX2 2016 Integration Marathon $Evaluation of $\int\frac{\sin (x+\alpha)}{\cos^3 x}\sqrt{\frac{\csc x+\sec x}{\csc x-\sec x}}dx$

Re: MX2 2016 Integration Marathon $Evaluation of $\int \frac{1-\cot^{9}x}{\tan x+\cot^{10}x}dx$ $Evaluation of $\int\frac{1}{1+x^6}dx$ $Evaluation of $\int\frac{\sqrt{\cot x}-\sqrt{\tan x}}{1+3\sin 2x}dx$

(1)$ Find the values of $a$ for which $ax^2+(a-3)x+1<0$ for$ $at least one positive $x$ (2)$ Find the values of $a$ for which $4^t-(a-4)2^t+\frac{9}{4}a<0\forall t \in (1,2)$

Re: MX2 2016 Integration Marathon $Let $I = \int\frac{\csc^2 x-2005}{\cos^{2005}x}dx = \int \frac{\csc^{2}x}{\cos^{2005}x}dx-\int\frac{2005}{\cos^{2005}}dx$ $Using Integration by parts for $(1)$ one $ I = -\frac{\cot x}{\cos^{2005}x}+\int 2005 \cos^{-2006}x\cdot \sin x\cdot \cot...

Re: MX2 2016 Integration Marathon $Evaluation of $\int\sqrt{\tan^2 x-2016}\;d\mathrn{x}$

Re: MX2 2016 Integration Marathon $Let $I = \int\frac{\cos 7x-\cos 8x}{1+2\cos 5x}dx = \int\frac{2\sin 5x \left(\cos 7x-\cos 8x\right)}{\cos 5x+\sin 10x}dx$ $Let $I = \int\frac{2\sin 5x \cdot \sin \frac{15x}{2}\sin \frac{x}{2}}{2\cos \frac{15x}{2}\cos \frac{5x}{2}}dx = \int 2\sin...

Re: MX2 2016 Integration Marathon $Let $I = \int\frac{2x^{12}+5x^9}{(x^5+x^3+1)^3}dx = \int\frac{2x^{12}+5x^9}{x^{15}\cdot \left(1+x^{-2}+x^{-5}\right)^3}$ $So $I = \int\frac{2x^{-3}+5x^{-6}}{\left(1+x^{-2}+x^{-5}\right)^3}dx$ $Now put $\left(1+x^{-2}+x^{-5}\right) = t\;,$ Then...

Re: MX2 2016 Integration Marathon But Paradoxica answer of my second Integral is \neq 0

$The length of latus rectum of parabola whose parametric coordinates$ $are $x=at^2+bt+c$ and $y=a't^2+b't+c'$

Re: MX2 2016 Integration Marathon $Evaluation of $\int_{0}^{\infty}\frac{\ln (x)}{x^2+9}dx$ $Evaluation of $\int_{-1}^{1}\cot^{-1}(\frac{1}{\sqrt{1-x^2}})\cot^{-1}(\frac{x}{\sqrt{1-(x^2)^{|x|}}})dx$ $Evaluation of $\int_{\frac{\pi}{4}}^{\frac{65\pi}{4}}\frac{1}{(1+2^{\cos x})(1+2^{\sin x})}dx$

$If $z_{1}\;,z_{2}\;,z_{3}\;,z_{4}$ are $4$ non zero complex number such $that $\Im(z_{1}+z_{2}) = \Im(z_{3}+z_{4}) =0\;,$ Then possible value of$ \arg(\frac{z_{1}}{z_{2}})+\arg(\frac{z_{3}}{z_{4}}) = \bf{Options::} (a)\;\; 0\;\;\;\;\;\; (b)\;\; \frac{\pi}{2}\;\;\;\;\;\; (c)\;\...

Yes InteGrand I mean same thing, You explained very well

$(1)\; Two point $P$ and $Q$ are taken on straight line $OA$ of length $a$ unit $ Then the probability that $PQ>b\;$ unit, Where $a>b$ $(2)\; The decimal part of logarithm of two numbers taken at random are$ $found to $7$ places , Then the probability that second number can be$...

Re: MX2 2016 Integration Marathon $Given $\displaystyle \int\frac{(x-1)\cdot \sqrt{x^4+2x^3+4x^2+2x+1}}{x^2(x+1)}dx = \int\frac{(x^2-1)\cdot \sqrt{x^4+2x^3+4x^2+2x+1}}{x^2(x^2+2x+1)}dx$ $Above we multiply both $\bf{N_{r}}$ and $\bf{D_{r}}$ by $(x+1).$ $So $\displaystyle =...