Tagged Questions

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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 ...
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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 ...
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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 ...
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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 ...
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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?
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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 ...
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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.
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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?
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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 $X$ ...
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 ...
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Logarithm checking my work.

$\ln e^{2x}=6$ 2x=6 x=3 Is this method correct. ln$e$ cancel into 1.
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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.. $$I=\frac{1}{2}\int_{0}^{\frac{\pi }{2}}\frac{\ln(\sin y)\ln(\cos y)}{\sin y\cos y}dy$$
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) • ...
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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 ...
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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.
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 ...
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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? $$\lim_{x\to 0}\frac{\log\cos x}{\log\cos 3x}$$
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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 ...
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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 ...
Baffling identity: $\prod \log_{10} \tan = \sum \log_{10} \tan$
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 ...