calculus - Evaluating $\int \frac {1} { {x^4+1}} dx$ - Mathematics ...
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 ...
Evaluating $\\sum_{k=0}^n\\binom\\alpha k^2\\lambda^k$
4 days ago · Is there a closed-form expression for the series $$ \\sum_{k=0}^n\\binom\\alpha k^2\\lambda^k,\\quad \\alpha ~ \\text{is non-integer} $$ There is an identity involving binomial …
Evaluating $\cos (i)$ - Mathematics Stack Exchange
Nov 27, 2020 · This is too long for a comment. So I will write it as an answer. Lets assume the definition of $\exp$ function via power series. Then it is well defined on the complex plane as well and Euler's …
Evaluating $\int_ {-\infty}^ {\infty} \frac {x^6} { (1 + x^4)^2} dx$
Oct 30, 2025 · I am currently stuck on this question and need some help in figuring out where my mistake is. Take complex function $f(z) = \\frac{z^6}{(1 + z^4)^2}$ and integrate ...
integration - Evaluating $\iiint z (x^2+y^2+z^2)^ {−3/2}\,dx\,dy\,dz ...
Jul 29, 2020 · Spherical Coordinate Homework Question Evaluate the triple integral of $f (x,y,z)=z (x^2+y^2+z^2)^ {−3/2}$ over the part of the ball $x^2+y^2+z^2\le 81$ defined by ...
integration - Evaluating $\sum_ {m=0}^\infty \sum_ {n=0}^\infty \frac ...
Nov 11, 2025 · I am evaluating the following integral: $$\\int_0^{1} \\left(\\tanh^{-1}(x) + \\tan^{-1}(x)\\right)^2 \\; dx$$ After using the Taylor series of the two functions, we ...
Evaluating $\\lim_{x\\to1}\\frac{m}{1-x^m} -\\frac{n}{1+x^n}$, for ...
Nov 11, 2025 · $$\\lim_{x\\to1}\\frac{m}{1-x^m} -\\frac{n}{1+x^n} \\;\\;\\;\\;\\;\\; m,n\\in \\mathbb{N}$$ My teacher had given the class this sum as homework. He gave us a hint ...
Evaluating $\\lim\\limits_{R\\to +∞}\\iint_{x^2+y^2\\leq R^2}\\left ...
Dec 20, 2021 · Now in order to prove that \begin {gather*} \small \lim_ {R → +∞} \iint\limits_ {\substack {1 \leqslant r \leqslant \sqrt {2} \\ 0 \leqslant \leqslant 2π}} \frac ...
Evaluating $\sum_ {i=1}^ {\infty}\frac { (i\ln 2)^i} {2^ii!}$
Dec 26, 2024 · I seek the proof of the evaluation to the sum $$\sum_ {i=1}^ {\infty}\frac { (i\ln 2)^i} {2^ii!} = \frac {1} {1-\ln2}-1 \approx 2.25889.$$ It is almost a power series ...
evaluating $\Gamma (1/2)$ using elementary methods
Aug 22, 2021 · evaluating $\Gamma (1/2)$ using elementary methods Ask Question Asked 4 years, 3 months ago Modified 4 years, 3 months ago