I am trying to evaluate the integral $$\int \frac {1} {1+x^4} \mathrm dx.$$ The integrand $\frac {1} {1+x^4}$ is a rational function (quotient of two polynomials), so I could solve the integral if I ...

Oct 17, 2017 · It is obvious that we should use Euler's formula, but the fact that $\\Vert e^{i \\alpha} \\Vert = 1$ (while the base is 2) brings difficulty of using it. Can anyone think of a way …

May 19, 2013 · Can this integral be solved with contour integral or by some application of residue theorem? ∫∞ 0 log(1 + x) 1 + x2 dx = π 4log2 + Catalan constant It has two poles at ± i and …

Evaluating integrals with sigma notation Ask Question Asked 13 years, 2 months ago Modified 8 years, 1 month ago

Jun 3, 2024 · Evaluating ∫2π 0 cos2(x) sin(x) dx ∫ 0 2 π cos 2 (x) sin (x) d x Ask Question Asked 1 year ago Modified 1 year ago

May 28, 2021 · I was following an explanation of the gamma function and everything made sense until the author evaluated the function at 1/2. The $\Gamma$-function is defined as the …

The important thing to know at this level of evaluating limits is that if the numerator is zero, you can only conclude the whole thing is zero if the denominator is not zero. We sometimes say 0 …

I have seen the Fresnel integral $$\int_0^\infty \sin x^2\, dx = \sqrt {\frac {\pi} {8}}$$ evaluated by contour integration and other complex analysis methods, and I have found these methods to …

Dec 12, 2017 · I saw this problem somewhere recently and I was having some difficulty getting started on it. The problem is twofold. The first is to evaluate: $$\sum_ {k=0}^\infty \left (\frac {1} …

I am trying to prove that \begin {equation} \int_ {0}^ {1}\frac {\log\left (x\right) \log\left (\, {1 - x^ {4}}\,\right)} {1 + x^ {2}} \,\mathrm {d}x = \frac {\pi^ {3 ...