1
vote
3answers
53 views

Unable to differentiate $\cos(x) \cos(2x) \cos(3x)$ and $\sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}$

I apologize for the lack of LaTeX. I will update this question with the proper LaTeX as soon as possible. I am having trouble with two differentiation exercise questions and was hoping someone could ...
3
votes
1answer
62 views

Finding $\int_{0}^{1} \frac{\log(1+x)}{1+x^2} {\rm d}x$ by differentiating under the integral sign.

I've tried to find this integral by the method already outlined in the title. I decided to let $$ \displaystyle I(\alpha) = \int_{0}^{1} \dfrac{\log(1+\alpha x)}{1+x^2} \text{ d}x. $$ From this ...
-1
votes
0answers
29 views

How can I count logarithm of sinus?

I've red a book when was the information about logarithm of sinus/cosine etc. There was description of the logarithm sinus/cosine tables and how to using that. But there wasn't description how to ...
1
vote
2answers
53 views

Algebraically, how are $-\ln|\csc x + \cot x| +C $ and $\ln| \csc x - \cot x|+C$ equal?

Algebraically, how are $-\ln|\csc x + \cot x| +C $ and $\ln| \csc x - \cot x|+C$ equal? I know both of these are the answer to $\int \csc x \space dx$, and I am able to work them out with calculus ...
3
votes
1answer
52 views

Rational values of $\sin(\log(x))$

Apart from the trivial solution $\sin(\log(1))=0$, is $$\sin(\log(x))$$ ever rational if $x$ is rational?
2
votes
1answer
85 views

How does the Sin and Cos scale on a slide rule work and what is the formula for it?

As I described in this question, I am trying to make a printable slide rule (similar to the slide rule provided in this SciAm article). I have made most of the parts of it but I can't make the Sin and ...
27
votes
0answers
298 views
+100

Ramanujan log-trigonometric integrals

I discovered the following conjectured identity numerically while studying a family of related integrals. Let's set $$ R^{+}:= \frac{2}{\pi}\int_{0}^{\pi/2}\sqrt[\normalsize{8}]{x^2 + \ln^2\!\cos x} ...
3
votes
3answers
61 views

Evaluate the limit of $\ln(\cos 2x)/\ln (\cos 3x)$ as $x\to 0$

Evaluate Limits $$\lim_{x\to 0}\frac{\ln(\cos(2x))}{\ln(\cos(3x))}$$ Method 1 :Using L'Hopital's Rule to Evaluate Limits (indicated by $\stackrel{LHR}{=}$. LHR stands for L'Hôpital Rule) ...
8
votes
2answers
200 views

A closed form for $\int_{0}^{\pi/2}\frac{\ln\cos x}{x}\mathrm{d}x$?

The following integrals are classic, initiated by L. Euler. \begin{align} \displaystyle \int_{0}^{\pi/2} x^3 \ln\cos x\:\mathrm{d}x & = -\frac{\pi^4}{64} \ln 2-\frac{3\pi^2}{16} ...
17
votes
4answers
655 views

A closed form for $\int_{0}^{\pi/2} x^3 \ln^3(2 \cos x)\:\mathrm{d}x$

We already know that \begin{align} \displaystyle & \int_{0}^{\pi/2} x \ln(2 \cos x)\:\mathrm{d}x = -\frac{7}{16} \zeta(3), \\\\ & \int_{0}^{\pi/2} x^2 \ln^2(2 \cos x)\:\mathrm{d}x = ...
1
vote
0answers
60 views

Logarithm and “basic” functions.

To express the antiderivatives of $\frac{1}{x}$, we cannot apply the formula $\int x^n dx=\frac{x^{n+1}}{n+1}+C$ and we need to introduce a new function, the logarithm. But how can we prove that ...
3
votes
4answers
80 views

Arctan equalling a logarithm?

Is $$\int \frac{dx}{ax^2+bx+c}=\frac{1}{a}\int{dx}{(x-\alpha)(x-\beta)}=\frac{1}{a} \int \left( \frac{ \frac{1}{\alpha-\beta} }{x-\alpha} - \frac{\frac{1}{\alpha-\beta}}{x-\beta}\right)=\frac{1}{a} ...
0
votes
1answer
40 views

How is $\arctan(\sinh(x * \pi))$ the inverse of $\log(\tan(x)) / \pi$

How is $$\arctan(\sinh(x * \pi))$$ the inverse of $$\frac{\log(\tan(x))}{\pi}$$ What is the relationship between $\log(x)$ and $\sinh(x)$. I guess is what my real question is.
1
vote
2answers
106 views

How can we relate calculus, trigonometry etc in real life

I have always wondered what does trigonometry, calculus, logarithms solve real world problems? Where do they apply in real life? Is there any simple book where I can understand it?
0
votes
1answer
45 views

Logarithms with trigonometric inequality

My class is going to have an exam tomorrow, but we can't figure out how to solve such equations. $$\log_{\ \large tg(x)} \sqrt{\sin(x)^2 - 5/12} < 1 $$ We tried to transform $1$ to $\log_{\ ...
0
votes
2answers
64 views

Are the sums equal to each other?

They are $2$ different results for the integral $$\int xe^{2x}\sin\left(\frac x3\right)\,dx$$ $\displaystyle\frac{-3}{1369}e^{2x}\left(3(35-74x)\sin\left(\frac x3\right)+(37x-36)\cos\left(\frac ...
0
votes
2answers
110 views

How do I solve this integral?

As stated the title, I get to a point which I can't do anything, and I'm sure I've made a mistake some where, here is my full working out: $$ \int e^{ix}\cos(x)dx \\ u = e^{ix} \text{ | } u'= ie^{ie} ...
2
votes
1answer
87 views

Can someone please simplify this, please.

After solving my previous question, click here for question page, I tried to go up a notch and complicate the question just a bit further, turns out $\int e^x\sin(x)\cos(x)dx$ is much more different ...
1
vote
3answers
116 views

Why $\int{\log^2(2\sin(\pi x))}dx\neq\frac{\log^3(2\sin(\pi x))\sin(\pi x)}{3\pi \cos(\pi x)}$?

Apparently I completely forgot the basics of calculus after 13 years of not studying it. Why won't: $$\frac{d \log^2(2\sin(\pi x))}{dx} = \frac{\log^3(2\sin(\pi x))\sin(\pi x)}{3\pi \cos(\pi x)}$$ ...
0
votes
3answers
257 views

Why is $-\ln(\cos(x))$ equal to $\ln(\sec(x))$?

Why does the value $-\ln(|\cos(x)|)$ become $\ln(|\sec(x)|)?$ I was doing an integral and I got my final answer as that, but I don't understand how you can just send the negative sign inside and make ...
18
votes
2answers
778 views

Integral $\int_0^{\pi/2}\frac{x}{\sin x}\log^2\left(\frac{1+\cos x-\sin x}{1+\cos x+\sin x}\right)dx$

Please help me to evaluate this integral: $$\large\int_0^{\pi/2}\frac{x}{\sin x}\log^2\left(\frac{1+\cos x-\sin x}{1+\cos x+\sin x}\right)dx$$
9
votes
1answer
148 views

An integral involving the inverse of $f(x)=\log x-\log\cos x+x\tan x$

Let the function $f:\left(0,\,\displaystyle\frac\pi2\right)\to\mathbb{R}$ be defined as $$f(x)=\log x-\log\cos x+x\tan x.$$ Let its inverse be denoted as ...
8
votes
1answer
222 views

Integral $\int_0^\infty\frac{dx}{\frac{x^4-1}{x\cos(\pi\ln x)+1}+2x^2+2}$

I need your help with this integral: $$\int_0^\infty\frac{dx}{\frac{x^4-1}{x\cos(\pi\ln x)+1}+2\,x^2+2}.$$ I wasn't able to evaluate it in a closed form, although an approximate numerical evaluation ...
1
vote
1answer
54 views

domain of function with logarithmic terms.

what will be the domain of function given below? $$y=1+3(\log(\sin(x))+\log(\csc(x)))$$ in book it is given this is valid for the values of angles of 1st and 2nd quadrant only. why this function is ...
0
votes
1answer
264 views

Evaluation of the integral $\int \cos\omega t\ln\cos\omega t\,dt$

I am trying to evaluate an integral of the form $$ \int \cos\left(\omega t\right) \ln \cos\left(\omega t\right) dt$$ and am unsure how to proceed. I rewrote it as: $$ \textrm{Re} \left\{\int dt ...
1
vote
0answers
86 views

simplification of a natural log of a trigonometric function

hope you are all well. I am having a bit of a mental block, I am wondering if it is possible to simplify the following expression: $$k\cos X \cdot 4\ln(\cos X)$$ where $k$ is a constant and $X$ ...
7
votes
3answers
353 views

Broken Calculator: only certain unary functions work.

I have run into a challenge on Codecademy.com that has me absolutely bewildered. I'm sure I'm just overlooking an obvious solution, but I've been scouring tables of trigonometric and logarithmic ...
0
votes
1answer
33 views

Logarithm checking my work.

$\ln e^{2x}=6$ 2x=6 x=3 Is this method correct. ln$e$ cancel into 1.
16
votes
4answers
468 views

How to calculate $I=\frac{1}{2}\int_{0}^{\frac{\pi }{2}}\frac{\ln(\sin y)\ln(\cos y)}{\sin y\cos y}dy$?

How do I integrate this guy? I've been stuck on this for hours.. $$I=\frac{1}{2}\int_{0}^{\frac{\pi }{2}}\frac{\ln(\sin y)\ln(\cos y)}{\sin y\cos y}dy$$
4
votes
2answers
314 views

Natural logarithm, equivalent function

I'm developing a software with a tool unable to "recognize" the ln(), so is there a way to get the equivalen to ln() using someones of functions below? • sin1(a) • cos1(a) • tan1(a) • ...
7
votes
3answers
744 views

Evaluating $\int_0^{\large\frac{\pi}{4}} \log\left( \cos x\right) \, \mathrm{d}x $

It's my first post here and I was wondering if someone could help me with evaluating the definite integral $$ \int_0^{\Large\frac{\pi}{4}} \log\left( \cos x\right) \, \mathrm{d}x $$ Thanks in ...
5
votes
3answers
635 views

Verifing $\int_0^{\pi}x\ln(\sin x)\,dx=-\ln(2){\pi}^2/2$

I used all I know to show that $$\int_0^\pi x\ln(\sin x)dx=-\ln(2) \pi^2/2$$ This is my homework but don't know where to start. I appreciate your help.
0
votes
1answer
358 views

Why does $\ln|\cot x|=\ln|\cos x|-\ln|\sin x|$ hold?

I am learning trigonometry. I can solve simple trigonometric equations. But its integration with log always confuses me. I am thinking about this sums since last two hours but can't find the ...
1
vote
4answers
170 views

How can I calculate $\lim_{x \to 0} \log(\cos(x))/\log(\cos(3x))$ without l'Hopital?

How can I calculate the following limit without using, as Wolfram Alpha does, without using l'Hôpital? $$ \lim_{x\to 0}\frac{\log\cos x}{\log\cos 3x} $$
5
votes
1answer
161 views

Why does $\cos (\pi\cos (\pi \cos (\log (20+\pi)))) \approx -1$

I read on Wikipedia that $$\cos (\pi\cos (\pi \cos (\log (20+\pi)))) \approx -1$$ to a high degree of accuracy. Why is this true? Is this pure coincidence or is there some mathematical ...
15
votes
3answers
2k views

How does e, or the exponential function, relate to rotation?

$e^{i \pi} = -1$. I get why this works from a sum-of-series perspective and from an integration perspective, as in I can evaluate the integrals and find this result. However, I don't understand it ...
2
votes
2answers
381 views

Baffling identity: $\prod \log_{10} \tan = \sum \log_{10} \tan$

I am quite a baffled now, I am not getting by how it can be written that : \begin{align*} &\log_{10} \tan 40^\circ \cdot \log_{10} \tan 41^\circ \cdot \log_{10} \tan 42^\circ \cdot\log_{10} \tan ...
10
votes
5answers
2k views

Understanding imaginary exponents

Greetings! I am trying to understand what it means to have an imaginary number in an exponent. What does $x^{i}$ where $x$ is real mean? I've read a few pages on this issue, and they all seem to ...