Issues with integral

[Albert Chan]

(08-19-2020 09:03 PM)Mordicus1973 Wrote:  int(1/(sqrt(cos(x))+sqrt(sin(x)))^4,x,0,pi/2)

XCas> gaussquad(1/(sqrt(cos(x))+sqrt(sin(x)))^4,x,0,pi/2)     → 0.333333333333

Doing integral symbolically, let t = tan(x) → dt = sec(x)^2 dx

\(\Large{1 \over (\sqrt{\cos x}+\sqrt{\sin x})^4} = {\sec^2 x \over (1+\sqrt{\tan x})^4} \)

XCas> F := int(1/(1+sqrt(t))^4, t)      → \(\Large
\frac{2 (-3\cdot \sqrt{t}-1)}{6 \left(\sqrt{t}+1\right)^{3}}\)

Limit of x = 0 to pi/2, same as t = 0 to +inf

XCas> preval(F,0, inf, t)                     → = 0 - (-1/3) = 1/3