Questions tagged [integration]
For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.
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Where does this 1 come from when balancing this integral equation?
$$ \int e^{ax}\cos(bx)\,\mathrm dx = \frac1{a}e^{ax}\cos(bx) + \frac{b}{a^2}e^{ax}\sin(bx) - \frac{b^2}{a^2}\int e^{ax}\cos(bx)\,\mathrm dx$$
$$\left(1 + \frac{b^2}{a^2}\right)\int e^{ax}\cos(bx)\,\...
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Evaluating $\int_0^4 \int_{\sqrt {x}}^2 \frac{x^2 e^{y^2}}{y^5}\mathrm dy\,\mathrm dx$
I need help starting with this. I can't find an example like this anywhere in my book
$$\int_0^4 \int_{\sqrt {x}}^2 \frac{x^2 e^{y^2}}{y^5}\mathrm dy\,\mathrm dx$$
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$\int \frac{dx}{(x^4 + 1)^2}$
What would be a relatively simple method for computing the indefinite integral below?
$\displaystyle \int \frac{dx}{(x^4+1)^2}$
Furthermore, how would one evaluate the following, possibly by ...
- 1,498
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Integral with spherical symmetry over cube
Is it possible to calculate the integral
$$I = \int_{-1}^1 \mathrm dx \int_{-1}^1 \mathrm dy \int_{-1}^1 \mathrm dz \frac{1}{x^2 + y^2 + z^2}$$
analytically? I tried using spherical coordinates
$$I ...
- 345
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Polar Coordinates and Double Integrals
Problem 1:
Find the area enclosed by the ellipse $\displaystyle \frac {1} {r} = 1 – 0.6 \cos(\theta)$.
We know $0\leq \theta\leq 2\pi$.
We know $0\leq r\leq 1/(1-0.6\cos(\theta))$.
Questions:
...
- 51
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Riemann-Stieltjes Integral
Define:
Partition $P = \{x_0,\dots,x_n\}$
$L(P) = \sum_{i=1}^n (\alpha(x_i)-\alpha(x_{i-1}) \inf (f(x): x \in [x_{i-1}, x_i])$
$U(P) = \sum_{i=1}^n (\alpha(x_i)-\alpha(x_{i-1}) \sup (f(x): x \in [...
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Showing that $\int_{-\infty}^{\infty} \exp(-x^2) \,\mathrm{d}x = \sqrt{\pi}$ [duplicate]
Possible Duplicate:
Proving $\\int_{0}^{+\\infty} e^{-x^2} dx = \\frac{\\sqrt \\pi}{2}$
The primitive of $f(x) = \exp(-x^2)$ has no analytical expression, even so, it is possible to evaluate $\...
- 2,328
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Problem calculating an integral over a surface
I've been trying to solve this for awhile and can't find a way.
Given $ S={(x,y,z) \in R^3 : z = x^2 - y^2 , x^2 + y^2 \leq 1 } $ and $\phi :R^3 \to R $ defined as $\phi (x,y,z)= (4z +8y^2 + 1)^{3/2}$,...
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Infinite area under a curve has finite volume of revolution?
So I was thinking about the harmonic series, and how it diverges, even though every subsequent term tends toward zero. That meant that its integral from 1 to infinity should also diverge, but would ...
- 1,606
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How much does symbolic integration mean to mathematics?
(Before reading, I apologize for my poor English ability.)
I have enjoyed calculating some symbolic integrals as a hobby, and this has been one of the main source of my interest towards the vast ...
- 170k
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Existence of an Analytic Formula for a Definite Integral
It would be very helpful if the following definite integral or a similar one had an analytic solution:
$$\int\nolimits_{-\infty}^{\infty}\mathrm{sech}^2(x)\exp(-\alpha x^2)\,\mathrm dx,\qquad \alpha \...
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Multivariable calculus double integral to polar coordinates
The task is to note down $\iint_D F(x,y)\mathrm dy\mathrm dx$ lane rows in polar coordinates. And region D is defined by $x^2 + y^2 = ax,\, a > 0 $ and $x^2 + y^2 = by,\, b > 0 $ intersection.
...
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Compute unknown limit from known integral in Mathematica?
I have an integral
$\int_a^b \! f(x) \, \mathrm{d}x = c$
where I know c and either a or b. Now I want to compute either b or a (i.e., the missing limit). How would I do that in Mathematica/Wolfram ...
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Expressing the given integral in terms of summation
Can anyone help me in changing the integral into the given form: $$\lim_{n \to \infty}n^{2} \Biggl(\ \ \int\limits_{0}^{1} \sqrt[n]{1+x^{n}} \ \text{dx}-1 \Biggr) = \sum\limits_{n=1}^{\infty} \frac{(-...
user9413
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Convergence of $\int_1^\infty \! \sin^\alpha(1/x) \, \mathrm{d}x$ for $\alpha > 1$
How do I show that for $\alpha > 1$ the integral $\displaystyle \int_1^\infty \! \sin^\alpha(1/x) \, \mathrm{d}x$ converges?
I am given the hint:
Compare with the integral $\displaystyle \...
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