Richard
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Interesting Integral including $\ln x$
2 votes

As you said, you're in high-school. So, the most obvious method is to use the substitution method and the make an observation that the resulting integral involves the Exponential Integral of complex ...

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How do I find the $n$th derivative of $x^n\ln\left(x\right)$?
1 votes

Leibnitz's theorem basically employs the binomial theorem with powers representing orders of derivatives. $\dfrac{d^n}{dx^n}\left(x^n\ln(x)\right)=\sum_{k=0}^{n}\binom{n}{k}(x^n)^{(n-k)}\ln^{(k)}(x).$ ...

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PMF from Ewen's Sampling Formula
0 votes

Find the marginal pmf of $m_i.$ Next, use the convolution theorem to find the pmf of $m_1+m_2,$ say, and see if this can be generalized to more than 2 random variables. Start and if you get stuck......

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Solve double integral using change of variables
0 votes

There is no way to represent $\int\cos(x^2)dx$ in terms of elementary functions! This is called the Fresnel integral, see Fresnel Integral. The best you can do is use series expansions.

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Integral of $\cos(\theta_1-\theta_2)/\sqrt{1-\cos(\theta_1-\theta_2)}$
0 votes

Mathematica says the integral converges to $(2\sqrt{2}\pi^2)i$. So it probably involves a complex substitution. I'm working on it. I will let you know if I figure it out.

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MLE for exponential distribution
0 votes

$\mathbb{I}(\cdot)$ is an indicator function I presume. In that case, your distribution does not have two parameters! It is only $\theta>0.$ Your distribution can actually be written in the ...

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