# Questions tagged [integration]

Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of the integrals being considered. This tag often goes along with the (calculus) tag.

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### Explain $-\int^{a}_{b}\frac{Q}{4 \pi \epsilon \bar{r}^{2}} \cdot d \bar{r}= \left[\frac{Q}{4\pi \epsilon \bar{r}}\right]^{a}_{b}$

Explain $-\int_{b}^{a}\frac{Q}{4 \pi \epsilon \bar{r}^{2}} \cdot d \bar{r}= \left[\frac{Q}{4\pi \epsilon \bar{r}}\right]_{b}^{a}$. What is the term $\frac{1}{\bar{r}^{2}} \cdot d\bar{r}$? I find it ...
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### lebesgue integral uniform convergence

Let $f_n, f : [a,b] \to R.$ If $f_n \to f$ uniformly then show that the lebesgue integrals are equal. ie. $\int f = lim \int f_n$ This is clearly true for continuous functions, but how do I ...
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### Is there a closed form for $\int x^n e^{cx}\,\mathrm dx$?

Wikipedia gives this evaluation: $$\int x^ne^{cx}\,\mathrm dx=\frac1cx^ne^{cx}-\frac nc\int x^{n-1}e^{cx}\,\mathrm dx=\left(\frac{\partial}{\partial c}\right)^n\frac{e^{cx}}{c}$$ But I have no idea ...
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### Proving that there exists a unique f(x) given Area and arc-length of f(x) on a given interval

I've been suggested to this site by some nice people at mathoverflow.net Before I get started, let me tell you a little about myself. I’m a fourth year Mechanical Engineering student at the ...
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### What's With The Integral $\int\sqrt{\cos(2\theta)}\, \mathrm d\theta$?

A student randomly asked me to compute $$\int\sqrt{\cos(2\theta)}\, \mathrm d\theta.$$ I was unable to do so, as were several other instructors. I typed the integral into Wolfram and it says that ...
### Integrating $\int\frac{2dx}{x\ln(6x)}$
I needed some help integrating this: $$\int\frac{2\,dx}{x\ln(6x)}.$$ I have never seen the dx within the problem like that, I am assuming I can't just move it to the outside can I? Can I start by ...