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help on definite integrals??

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Hello! Would anyone be able to help me evaluate the definite integral of this problem:
2 0[1/((sqrt1+x^3)) dx

Thank you!

Mar 23, 2019

#1
+5172
+2

$$\displaystyle \int_0^2 \dfrac{1}{\sqrt{1+x^3}}~dx$$

is that what you mean?

Mar 23, 2019
#2
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yes!!!

Guest Mar 23, 2019
#3
+5172
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It's a mess...

$$\displaystyle \int \dfrac{1}{\sqrt{1+x^3}}~dx = \\ \dfrac{2 \sqrt[6]{-1} \sqrt{-\sqrt[6]{-1} \left(x+(-1)^{2/3}\right)} \sqrt{(-1)^{2/3} x^2+\sqrt[3]{-1} x+1} F\left(\sin ^{-1}\left(\frac{\sqrt{-(-1)^{5/6} (x+1)}}{\sqrt[4]{3}}\right)|\sqrt[3]{-1}\right)}{\sqrt[4]{3} \sqrt{x^3+1}}$$

$$\displaystyle \int_0^2 \dfrac{1}{\sqrt{1+x^3}}~dx =2 \cdot \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};-8\right) \approx 1.40218$$

http://functions.wolfram.com/EllipticIntegrals/EllipticF/

http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/

Rom  Mar 23, 2019