Find the integral of sin^2010 x/ 2010

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Homework Statement



\int\frac{sin^{2010}x}{2010}



The Attempt at a Solution



All that I could do was to :
\frac{1}{2010^{2}}\int tan xdt

Please help me futher!
 
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t=sin^{-1}t^{\frac{1}{2010}}
 
Just to clarify, you meant \int \frac{sin^{2010}x}{2010} dx right?

Are you sure this isn't supposed to be a definite integral? Because otherwise the indefinite integral would take about forever to evaluate.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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