Evaluating $\\lim_{n\\to\\infty} \\int_1^\\infty …
6 days ago · $$ I_n=\int_ {1}^ {\infty}\frac {n\bigl (x^ {\alpha+1}-x^\alpha\bigr)\sin\!\left (\frac {1} {x}-1\right)} {x^3\bigl (x^\alpha+n^\alpha (x-1)^\alpha\bigr)}\,dx, \qquad ...
integration - Evaluating $ \int_ {1/2}^ {\infty} \frac {\Gamma (u ...
Dec 13, 2025 · Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and …
calculus - Evaluating $\int \frac {1} { {x^4+1}} dx$ - Mathematics ...
I am trying to evaluate the integral $$\int \frac {1} {1+x^4} \mathrm dx.$$ The integrand $\frac {1} {1+x^4}$ is a rational function (quotient of two polynomials), so I could solve the integral if I ...
Evaluating $\\sum_{k=0}^n\\binom\\alpha k^2\\lambda^k$
Dec 9, 2025 · Is there a closed-form expression for the series $$ \\sum_{k=0}^n\\binom\\alpha k^2\\lambda^k,\\quad \\alpha ~ \\text{is non-integer} $$ There is an identity involving binomial …
calculus - Evaluating $\int {\frac {x^ {14}+x^ {11}+x^5}
Jul 2, 2025 · The following question is taken from JEE practice set. Evaluate $\displaystyle\int {\frac {x^ {14}+x^ {11}+x^5} {\left (x^6+x^3+1\right)^3}} \, \mathrm dx$. My ...
Evaluating $ \\lim_{x \\to 0} \\frac{e - (1 + 2x)^{1/2x}}{x} $ without ...
Sep 11, 2024 · The following is a question from the Joint Entrance Examination (Main) from the 09 April 2024 evening shift: $$ \lim_ {x \to 0} \frac {e - (1 + 2x)^ {1/2x}} {x} $$ is equal to: (A) …
calculus - Evaluating $I=\int_ {0}^ {\frac {\pi} {2}}\prod_ {k=1}^ {7 ...
Oct 23, 2024 · I am attemping to show that $$ I \equiv \int_ {0}^ {\pi/2}\left [\prod_ {k = 1}^ {7}\cos\left (kx\right)\right] {\rm d}x = \frac {\pi} {32} $$ So far I have tried ...
Evaluating $\int \cot x \csc^2x \,\mathrm {d}x$ with $u=\cot x$
Also, $\mathrm d\cot (x) = -\csc^2 (x) \mathrm dx$, not $-\cot (x) \csc (x)\mathrm dx$.
integration - Evaluating $\iiint z (x^2+y^2+z^2)
Jul 29, 2020 · Spherical Coordinate Homework Question Evaluate the triple integral of $f (x,y,z)=z (x^2+y^2+z^2)^ {−3/2}$ over the part of the ball $x^2+y^2+z^2\le 81$ defined by ...
Evaluating $\\prod_{n=1}^{\\infty}\\left(1+\\frac{1}{2^n}\\right)$
Sep 13, 2016 · Compute:$$\prod_ {n=1}^ {\infty}\left (1+\frac {1} {2^n}\right)$$ I and my friend came across this product. Is the product till infinity equal to $1$? If no, what is the answer?