Integral (maybe simple maybe hard)

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Homework Statement


integrate (t^2) / (1+2t) . Wolfram alpha gave this as the answer: http://www4a.wolframalpha.com/Calculate/MSP/MSP957620di145hih25dggd00002hc5g230hb2h25hd?MSPStoreType=image/gif&s=24&w=229.&h=36.

The Attempt at a Solution



I tried a u-substitution and couldn't arrive at a solution. Any other suggestions! Int by parts doesn't seem to work either!
 
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The only easy method that comes to mind for me is to split the integrand into easier parts by long division.
 
hello.world said:

Homework Statement


integrate (t^2) / (1+2t) . Wolfram alpha gave this as the answer: http://www4a.wolframalpha.com/Calculate/MSP/MSP957620di145hih25dggd00002hc5g230hb2h25hd?MSPStoreType=image/gif&s=24&w=229.&h=36.

The Attempt at a Solution



I tried a u-substitution and couldn't arrive at a solution. Any other suggestions! Int by parts doesn't seem to work either!
The substitution u = 1+2t should work. I'm guessing that's what you tried, but you messed up somewhere along the way. Show your work.
 
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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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