How to choose between the principal root (complex) and the real root when calculating a definite integral?

Question

The following integral gives a complex answer in Wolfram Cloud (basic account):

N[Integrate[(2+x)^(1/5)*(1+x^2)^(1/4),{x,-3,-0}]]

3.37527+0.809411i

However, Wolfram Alpha let me choose between using the real root or the principal root. The answer for the integral using the real root is 0.884167

I have tried with Assumptions->x>-2, Assumptions->x^2+1>0 but the answer remains complex. How can I obtain the real-valued solution for the integral in Wolfram Language?

Thanks.

In principle integration has nothing to add with respect to an old question Finding real roots of negative numbers (for example, $\sqrt[3]{-8}$) — Artes, Jul 08 '23 at 11:59

score 3 · Accepted Answer · answered Jul 08 '23 at 08:58

3

r=Integrate[Surd[(2+x),5]*(1+x^2)^(1/4),{x,-3,0}]
N[r]

Mathematica graphics

Chop[%]

Mathematica graphics

answered Jul 08 '23 at 08:58

Nasser

143,286
11
154
359

How to choose between the principal root (complex) and the real root when calculating a definite integral?

1 Answers1