contest math - Evaluating $4^ {5 \log_ {4\sqrt {2}} (3-\sqrt {6}) - 6 ...
May 9, 2025 · Here's another, seemingly monstrous question from a JEE Advanced preparation book. Evaluate the following expression: $$4^ {5 \log_ {4\sqrt {2}} (3-\sqrt {6}) - 6\log ...
Evaluating $\\lim_{n\\to\\infty} \\int_1^\\infty …
5 days ago · $$ 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 ...
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 …
algebra precalculus - Evaluating $\frac {1} {a^ {2025}}+\frac {1} {b ...
Feb 21, 2025 · Well, the image equation is a different equation? One has $\frac1 {2024}$ on the right, and the other has $2024$ on the right?
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 ...
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) …
Evaluating $\lim\limits_ {n\to\infty} e^ {-n} \sum\limits_ {k=0}^ {n ...
I'm supposed to calculate: $$\\lim_{n\\to\\infty} e^{-n} \\sum_{k=0}^{n} \\frac{n^k}{k!}$$ By using WolframAlpha, I might guess that the limit is $\\frac{1}{2 ...
Evaluating $\\lim_{n\\to\\infty}\\left( \\frac{\\cos\\frac{\\pi}{2n ...
Jan 24, 2025 · Since the OP solve his/her problem, I just as well complete the solution: \begin {align} \frac {1} {n+1}\sum^n_ {k=1}\cos\left (\tfrac {k\pi} {2n}\right)&=\frac {n ...
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 ...