Integrate X^3sin(x^2) | No Substitutions, No Chaos

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Find the integral of

X^3sin(x^2)

No useful substitutions I can find, integration by parts gets chaotic.
 
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Write it as x(x2)sin(x2) and begin with substitution u = x2.
 
Got the answer. Thank you.

I wish I were able to recognize these things more quickly.
 
1MileCrash said:
Got the answer. Thank you.

I wish I were able to recognize these things more quickly.
Practice, practice, practice ...
 

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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