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I'm having trouble with this integral, I tried different t substition, u substition, integration by parts, but it just complicates it and I get nowhere. $$\int_{0}^{1}\sqrt[3]{x^2+1}dx$$

Catalin Toba
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  • Precisely which substitutions did you try? It's better if you're more specific about what you've tried, so we can better help you. – Christian E. Ramirez Jan 17 '23 at 21:45
  • There is no elementary antiderivative according to Wolfram Alpha, so you're going to have to apply some advanced trick that only works for definite integrals. – PrincessEev Jan 17 '23 at 21:46
  • My best guess at the moment for something approaching an answer would be a hyperbolic trig sub. Let $x = \sinh(u)$. Then $x^2+1 = \sin^2(u) + 1 = \cosh^2(u)$ and $dx = \cosh(u) , du$, so

    $$\int_0^1 \sqrt[3]{x^2+1} , dx = \int_0^{\sinh(1)} \cosh^{5/3}(u) , du$$

    An approximation could be made using the definition of $\cosh(x)$ in terms of $e^x$ and binomial series or other such items.

     – PrincessEev Jan 17 '23 at 21:51
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    Another post on it being nonelementary, and a series solution. – PrincessEev Jan 17 '23 at 21:53
  • Mathematica gives it as HypergeometricFunction2F1$[-1/6,1/2,4/2,-1]$ – Z Ahmed Jan 18 '23 at 01:57

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