Evaluating $\\int_1^{\\sqrt{2}} \\frac{\\arctan(\\sqrt{2-x^2})}{1+x^2 ...
Dec 30, 2025 · I was recently trying to compute the value of the integral $$\int_1^ {\sqrt {2}} \frac {\arctan (\sqrt {2-x^2})} {1+x^2}\,\mathrm dx.$$ I’ve tried differentiation under the integral sign, contour integra...
calculus - Evaluating $\int_0^1 \frac {\ln (1+x)} { (1+x) (1+x^2 ...
Mar 23, 2025 · Partial fraction decomposition of the integral would lead to, $$\begin {align}\int_0^1 \frac {\ln (1+x)} { (1+x) (1+x^2)} \, dx& = \frac {1} {2}\int_0^1\frac {\ln (1 ...
contest math - Evaluating $4^ {5 \log_ {4\sqrt {2}} (3-\sqrt {6}) - 6 ...
May 9, 2025 · Here's another, seemingly monstrous question from a JEE Advanced preparation book. Evaluate the following expression: $$4^ {5 \log_ {4\sqrt {2}} (3-\sqrt {6}) - 6\log ...
Evaluating $ \\lim_{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) $0$ (B) $\frac { …
calculus - Evaluating $\int {\frac {x^ {14}+x^ {11}+x^5} { (x^6+x^3+1 ...
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 ...
algebra precalculus - Evaluating $\frac {1} {a^ {2025}}+\frac {1} {b ...
When I tried to solve this problem, I found a solution (official) video on YouTube. That is a = −b, c = 2024 a = b, c = 2024 and the correct answer is 1 20242025 1 2024 2025. Is there an alternative …
calculus - Evaluating $\int \frac {1} { {x^4+1}} dx$ - Mathematics ...
The integrand 1 1+x4 1 1 + x 4 is a rational function (quotient of two polynomials), so I could solve the integral if I can find the partial fraction of 1 1+x4 1 1 + x 4. But I failed to factorize 1 +x4 1 + x 4. Any …
Evaluating $\\lim_{n\\to\\infty}\\left( \\frac{\\cos\\frac{\\pi}{2n ...
Jan 24, 2025 · The problem is to solve: $$\lim_ {n\to\infty}\left ( \frac {\cos\frac {\pi} {2n}} {n+1}+\frac {\cos\frac {2\pi} {2n}} {n+1/2}+\dots+\frac {\cos\frac {n\pi} {2n}} {n+1 ...
integration - Evaluating $\iiint z (x^2+y^2+z^2)^ {−3/2}\,dx\,dy\,dz ...
Jul 29, 2020 · Spherical Coordinate Homework Question Evaluate the triple integral of $f (x,y,z)=z (x^2+y^2+z^2)^ {−3/2}$ over the part of the ball $x^2+y^2+z^2\le 81$ defined by ...
Evaluating $\int_ {0}^1\int_ {0}^1 xy\sqrt {x^2+y^2}\,dy\,dx$
Calculate the iterated integral: $$\int_ {0}^1\int_ {0}^1 xy\sqrt {x^2+y^2}\,dy\,dx$$ I'm stumped with this problem. Should I do integration by parts with both variables or is there another way to do ...