Question

If $\underset{\pi /4}{\overset{\pi /3}{\int }}\frac{({\sin }^3\theta -{\cos }^3\theta -{\cos }^2\theta )(\sin \theta +\cos \theta +{\cos }^2\theta {)}^{2010}}{\left(\sin \theta {)}^{2012}\right(\cos \theta {)}^{2012}}d\theta =\frac{(a+\sqrt{b}{)}^n-(1+\sqrt{c}{)}^n}{d}$
where $a,b,c$ and $d$ are all positive integers, then the value of $a+b+c+d$ is

Answer 1

$I=\underset{\pi /4}{\overset{\pi /3}{\int }}\frac{({\sin }^3\theta -{\cos }^3\theta -{\cos }^2\theta )(\sin \theta +\cos \theta +{\cos }^2\theta {)}^{2010}}{\left(\sin \theta {)}^{2012}\right(\cos \theta {)}^{2012}}d\theta$
$I=\underset{\pi /4}{\overset{\pi /3}{\int }}\frac{({\sin }^3\theta -{\cos }^3\theta -{\cos }^2\theta )(\sin \theta +\cos \theta +{\cos }^2\theta {)}^{2010}}{\left({\sin }^2\theta \right)\left({\cos }^2\theta \right)\left(\sin \theta {)}^{2010}\right(\cos \theta {)}^{2010}}d\theta$
$=\underset{\pi /4}{\overset{\pi /3}{\int }}(\tan \theta \sec \theta -\cot \theta \text{cosec}\theta -{\text{cosec}}^2\theta )(\sec \theta +\text{cosec}\theta +\cot \theta {)}^{2010}d\theta$

Put $\sec \theta +\text{cosec}\theta +\cot \theta =t$
$\Rightarrow (\tan \theta \sec \theta -\cot \theta \text{cosec}\theta -{\text{cosec}}^2\theta )d\theta =dt$
$\int t^{2010}dt=\frac{1}{2011}{t}^{2011}$
$\therefore I=\frac{1}{2011}[{(2+\frac{2}{\sqrt{3}}+\frac{1}{\sqrt{3}})}^{2011}-(\sqrt{2}+\sqrt{2}+1{)}^{2011}]$
$=\frac{1}{2011}\left[\right(2+\sqrt{3}{)}^{2011}-(1+\sqrt{8}{)}^{2011}]$
Hence, $a=2,b=3,c=8$ and $d=2011$
$\Rightarrow a+b+c+d=2024$