# Tagged Questions

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### Can't find solution to Calculus 8th (Adams, Essex) problem

I've been sitting here for hours trying to find a solution to his problem. If you have the function $g(y)$, which is the inverse of $f(x) = x^x,\\ e^{-1} \leq x < \infty,$ show that ...
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### ${\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$ and $\int_{0}^{\pi/2} \frac{\log \cos x}{x^2}\:\mathrm{d}x$

I have found the following new result connecting to rational log-cosine integrals. Proposition. \begin{align} \displaystyle & {\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} ...
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### Unable to comprehend a connection between two equations

I was reading this paper and got stuck at the transition from Equation (13) to Equation (14) (p. 16/17). We got a function of the form: $y(t)=k(t)^{\alpha}h(t)^{\beta}$ We know it grows from zero ...
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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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### Hints on calculating the integral $\int_0^1\frac{x^{19}-1}{\ln x}\,dx$

I would be happy to get some hints on the following integral: $$\int_0^1\frac{x^{19}-1}{\ln x}\,dx$$
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### Finding time constants of a circuit?

So this is a homework question and I am having trouble figuring out what they are asking. 'The potential difference (voltage) across the capacitor at time t > 0 is given by $V_C(t) = q(t)/C$. The ...
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### Summation of a function with the variable both in the function amd in the upper limit

E is defined as : E = c1 ( a$\rho$ + b$\rho ^{2}$ ) + c2 $\rho$ ( c + d $\sum_{j=0}^{n} (\log{ \frac{R\rho}{j} } )$ ) + c3 $\rho ^{2}$ a, b, c, d, c1, c2, c3, R are known constants. $\rho$ is the ...
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### Separating the log of a sum

I know there is no formula to separate the log of a sum, e.g. $\log(X+Y)$ into two parts, but are there any approximation rules that can allow me to achieve this objective? ...
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### Prove this logarithm equation

I keep getting the wrong answer. Can someone please correct my working out a^x=b^(1-x) In(a)^x=In(b)^(1-x) xIn(a)=(1-x)In(b) xIn(a)=In(e)-xIn(b) xIn(a)+xIn(b)=In(e) x[In(a)+In(b)]=Ine ...
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### Limit of a Logarithm with Different Bases

We are to compute $$\lim_{n->\infty}{\frac{2^{\log_3 n}}{3^{\log_2 n}}}$$ Clearly the bases are reversed between the logarithm and exponents, so I can't seem to find any logarithm or exponential ...
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### Inverse Function of Logarithm

The answer is A but I don't understand why! $-2 \log_e (x^2)$ can be re-written as $-4 \log_e(x)$ right? but why do these two graphs look different? the graph $-2 \log_e (x^2)$ is one to ...
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### Maximum value of constant in logarithm problem

The first thing I did was: make: (x-1)^2 - k > 0 (x-1)^2 > k don't know what to do after this point... the maximum value of k is 9 i dont really understand what the maximum value of k is? ...
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### Evaluate the integral. $\int x^2 \log(4x) dx$

The problem is $\int x^2 \log(4x) dx$ Here $\ln$ refers to the natural logarithm. So far, I know $u = x^2$ and $du = 2x (dx)$. So $dv = \ln(4x) dx$ and $v = 1/x$, but I don't know where to go from ...
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