# Tagged Questions

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### Volumes of Revolution Washer Method

I have to find the volume of revolution of a region called $C$ using around the $y=-1$ axis. The region is bounded above by $y \ = \ \ln(x+1)$, bounded below by $y=e^{-x}$ and on the right by $x=3$. ...
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### Arc Length in two dimensions by integration

I'm really at the end of my wits on this problem. Basically I'm trying to find arc length. The vector-valued function is: $R=\langle t,\sqrt{t}\rangle$ and $t\ge0$. We're looking for the length of ...
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### How to calculate integral $I=\displaystyle\int_{-1}^{1}\dfrac{dz}{\sqrt[3]{(1-z)(1+z)^2}}$?

The integral is $I=\displaystyle\int_{-1}^{1}\dfrac{dz}{\sqrt[3]{(1-z)(1+z)^2}}$. I used Mathematica to calculate, the result was $\dfrac{2\pi}{\sqrt{3}}$, I think it may help.
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### Fourier transform of t*(sent/pi*t)^2

Here's the function (I need it's fourier transform).
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### Solve the integeral equation (C.S.I.R)

Let $\lambda_1, \lambda_2$ be the eigen value and $f_1 , f_2$ be the coressponding eigen functions for the homogeneous integeral equation $$\phi(x) - \lambda \int_0^1 (xt +2x^2) \phi(t) dt = 0$$ ...
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### How do I evaluate the integral $\int_0^{\infty}\frac{x^5\sin(x)}{(1+x^2)^3}dx$?

I have no idea how to start, it looks like integration by parts won't work. $$\int_0^{\infty}\frac{x^5\sin(x)}{(1+x^2)^3}dx$$ If someone could shed some light on this I'd be very thankful.
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### physics math problem with dirac delta

I'm trying to do a homework problem where they have asked me to recover the usual coordinate space momentum operator from its hilbert space equivalent. it gets you to show: ...
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### How to evaluate this triple integral?

How would I go about evaluating this integral? I want to change the order of integration but don't know how. $$\int_0^1\int_1^{\Large e^z}\int_0^{\log y}x\ dx\,dy\,dz$$ I'm having difficulty ...
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### Evaluate integral

How do I evaluate the following integral, the answer according to Wolfram Alpha is $2$, but I keep on getting $0$ after using integration by parts.$$\frac12\int_{-\infty}^\infty x^2e^{-|x|}\ dx$$
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### How does one calculate $\int_0^1 \frac {\arcsin(x)}{x}dx$? [duplicate]

How can I evaluate the following? $$\int_0^1 \frac {\arcsin(x)}{x}dx.$$ Could not find a primitive, so I went for some other methods like arranging it as a double integral or introducing a ...
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### Help my homework: $\int_0^1\int_{0}^1\frac{x^2-y^2}{(x^2+y^2)^2}dy\, dx$ [duplicate]

I am trying to integrate $$\int_0^1\int_{0}^1\frac{x^2-y^2}{(x^2+y^2)^2}dy\, dx$$ In my book said that use tangent function but I don`t know how to evaluate it. Please help me. I want to know the ...
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### Calculate $\int_0^1 e^x dx$ as a limit of a sum?

As for now, I've been doing the opposite thing. For a given sum in terms of $n\in\mathbb{N}$ I had to calculate the limit (as $n$ approaches infinity) of that sum by applying: ...
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### Proof of integral equality

Let $f^{(n)}(x)$ be the $n$-th derivative of $f(x) = \cos(x)$. Prove that : $$\int_0^{2\pi} f^{(n)}(x) \,\, dx = \int_0^{2\pi} f^{(n)}(kx) \,\, dx,$$ where $n$, $k$ are natural numbers equal or ...
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### Area under the curve $f(x) = \sin x$

Find the area under the curve $f(x) = \sin x$ on the interval $[0, \pi]$ if $\sin x \ge 0$ My handbook give this as $$\int_0^\pi \sin x \space dx = (\cos \pi) - (\cos 0) = (-1) - (-1) = 2$$ what ...
I need to find a continuous and bounded function $\mathrm{f}(x)$ such that the limit $$\lim_{T\to\infty} \frac{1}{T}\, \int_0^T \mathrm{f}(x)~\mathrm{d}x$$ doesn't exist. I thought about ...