All Questions
7
questions
4
votes
0
answers
135
views
Simplify a summation in the solution of $\displaystyle\int_{0}^{\infty}e^{-cx}x^{n}\arctan(ax)\mathrm{d}x$
Context
I calculated this integral:
$$\begin{array}{l}
\displaystyle\int_{0}^{\infty}e^{-cx}x^{n}\arctan(ax)\mathrm{d}x=\\
\displaystyle\frac{n!}{c^{n+1}}\left\lbrace\sum_{k=0}^{n}\left[\text{Ci}\left(...
0
votes
0
answers
95
views
Proof of $sin$ formula.
I am reading this quesiton and accepted answer.
Question is about proof.
$S = \sin{(a)} + \sin{(a+d)} + \cdots + \sin{(a+nd)}$
$S \times \sin\Bigl(\frac{d}{2}\Bigr) = \sin{(a)}\sin\Bigl(\frac{d}{2}\...
- 879
1
vote
1
answer
208
views
Definite integral as limit of sum $\int_{a}^{b}\sin(x)dx$
I learn calculus and get stuck. I need help
We have $\int_{a}^{b}\sin(x)dx$, need to calculate using integral sums, so we split $[a,b]$ into $n$ equal parts: $dx \to h=\frac{b-a}{n}$. So
$$\sigma_{n} =...
- 11
0
votes
2
answers
80
views
How do I evaluate this finite sum using simple techniques?
I am trying to calculate the definitive integral by definition (with Riemann sum).
$$\int_{\frac{-\pi}{2}}^{\frac{3\pi}{2}} (2\sin{(2x+\frac{3\pi}{2})}) \ dx$$
But during the process of calculating ...
- 1,798
1
vote
2
answers
84
views
Is the value of $\sum_{x=0}^{\infty}\frac{\cos(\pi x)}{x!}=1/e$?
What is the value of $$\sum_{x=0}^{\infty}\frac{\cos (\pi x)}{x!}$$ I wrote $\cos (\pi x)=R (e^{ix}).e^{\pi} $ but the $x! $ is a trouble can someone help me out. And if value isnt $1/e $ then can we ...
- 16k
3
votes
3
answers
182
views
Calculate $\int_0^{1/10}\sum_{k=0}^9 \frac{1}{\sqrt{1+(x+\frac{k}{10})^2}}dx$
How can we evaluate the following integral:
$$\int_0^{1/10}\sum_{k=0}^9 \frac{1}{\sqrt{1+(x+\frac{k}{10})^2}}dx$$
I know basically how to calculate by using the substitution $x=\tan{\theta}$ :
...
- 109
0
votes
3
answers
405
views
Could we solve $\int_{0}^{\infty}\sin(x)dx$ and what does it say about $\lim_{x\to\infty}\cos(x)$?
As the title states: Could we solve $\int_{0}^{\infty}\sin(x)dx$ and what does it say about $\lim_{x\to\infty}\cos(x)$?
It is clear we can't solve this using the fundamental theorem of Calculus, but ...
- 74.9k