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whatamidoing
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8 votes
2 answers
304 views

How to solve the following limit

6 votes
2 answers
313 views

Ways to tackle the integral $\int_{0}^{\frac{\pi}{4}}\operatorname{Li}_3(\tan^4 x) \, dx$

6 votes
1 answer
192 views

Compute ; $\Gamma=\lim_{n\to\infty} \int_{t_n}^{t_{n+1}} \frac{(f(x-t_n))^{g(t_{n+1}-x)}}{(f(t_{n+1}-x))^{g(x-t_n)}+(f(x-t_n))^{g(t_{n+1}-x)}} \, dx$

5 votes
3 answers
344 views

How can $\int_0^1 \lfloor ax^2+bx+c \rfloor \,\Bbb dx$ be generalized?

5 votes
2 answers
169 views

Finding $k$ such that $\int_0^{\pi/12} \ln(\tan (3x))\,\mathrm{d}x = k \int_0^{\pi/12} \ln(\tan(x))\,\mathrm{d}x$

4 votes
3 answers
298 views

Use of $\sum_{r=1}^{\infty} \frac{1}{r^2} = \frac{\pi^2}{6} $ in evaluating $I$

4 votes
2 answers
165 views

Prove the following from $L = \int_0^1 x^2 \sin^ {-1 }(\frac{x^2}{1+\sqrt{1+x^4}}) \,dx$

4 votes
1 answer
108 views

If $x,y∈(-π,π]$, then find the area of the polygon formed by points $(x,y)$ satisfying the equation $\lfloor|\sin x|\rfloor+\lfloor|\cos y|\rfloor=2$.

4 votes
3 answers
178 views

Evaluating $J=\int_0^{\infty} \frac{x^3 \ln \left(e^x+\frac{x^3}{6}+\frac{x^2}{2}+x+1\right)-x^4}{\frac{x^3}{6}+\frac{x^2}{2}+x+1}\,dx$

4 votes
2 answers
2k views

Let A and B are square matrices of order 2 with real elements such that AB = $A^2B^2 - (AB)^2$

4 votes
5 answers
302 views

More methods to evalute this integral ; $I=\int_0^2 x(8-x^3)^{\frac{1}{3}} \, dx$

4 votes
1 answer
56 views

Find $f(n,r)$ in $\int \frac{1}{\prod_{r=0}^{n} (x+r)}\, dx=\sum_{r=0}^n \frac{(-1)^r}{f(n,r)}p(x)+K$

3 votes
1 answer
75 views

Show $ \lim_{x \to 0}\left[\int_0^{1} by+a(1-y)^x\,dy \right]^\frac{1}{x} = \frac{1}{e}\left[\frac{b^b}{a^a}\right]^\frac{1}{b-a}$ for $a < b$

3 votes
1 answer
210 views

Let $G$ be a circle of radius $R > 0$. Let $G_1, G_2,...,G_n$ be $n$ circles of equal radius $r > 0$, Then prove the following;

3 votes
1 answer
61 views

Use of Integration in proving the given relation with $\displaystyle \sum_{n=1}^{2022}$ $\frac{(-1)^{r-1}}{r}$ $\binom{2022}{r}$ as given; [duplicate]

2 votes
1 answer
69 views

How to solve the given differential equation $xg(f(x))f'(g(x))g'(x)= f(g(x))g'(f(x))f'(x)$?

2 votes
1 answer
87 views

Guidance to evaluating Integration of Greatest Integer function / Floor function

2 votes
1 answer
81 views

Evaluate the integral;

2 votes
1 answer
85 views

How to evaluate the given polynomials?

2 votes
1 answer
134 views

Evaluate $I = \int_0^{10} \lfloor x \rfloor^3 \{x\}\, dx$

2 votes
2 answers
79 views

How to evaluate series whose terms are constructed from terms in the Fibonacci sequence

1 vote
2 answers
84 views

Let L = $\frac{f^{2n-2}(0)}{\left(2n-2\right)!}$ and Let M = $\sum_{n=1}^{102}(-1)^{n-1}(L)$. Evaluate M.

1 vote
3 answers
105 views

Maximum of $G=1+k\cos \theta+k^2\cos(2\theta)+k^3\cos(3\theta)+\cdots,$

1 vote
1 answer
66 views

Evaluating $S(k)=\sum_{n=1}^\infty \frac{1}{\sum_{r=1}^n(r)^k}$ for $k>0$

1 vote
1 answer
117 views

$\frac{3}{2}\int_a^b \frac{1}{\sqrt[3]{1-\cos4x}}dx>\cot2a \, +\, \cot2b$

1 vote
3 answers
127 views

How to find the equation of an ellipse using three points?

1 vote
1 answer
52 views

How to maximize the area of the triangle?

1 vote
1 answer
88 views

$T=\lim_{n \to \infty} \int_{\frac{1}{\sqrt [n+1] {n!}}}^{\frac{1}{\sqrt [n+1]{(n-1)!}}} \frac{n+1}{x(n+1)ln(x)+xln((n+1)!)} \,dx$

0 votes
2 answers
98 views

What is the generalized solution to $M = \int_0^1 (1-x^m)^n$?

0 votes
2 answers
93 views

Evaluating this generalized integral