What Are the Different Solutions for Integrating xsin^2xdx?

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<br /> \int xsin^2xdx<br />
<br /> \frac{1}{2}\int x(1-cos2x)dx<br />
<br /> \frac{x^2}{4}-\frac{1}{2}\int xcos2xdx<br />
<br /> \frac{x^2}{4}-\frac{xsin2x}{4}-\frac{cos2x}{8}+C<br />
 
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diffrent solutions
 
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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