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- What is the integral of 1 x? - Mathematics Stack Exchange
16 Answers to the question of the integral of 1 x 1 x are all based on an implicit assumption that the upper and lower limits of the integral are both positive real numbers If we allow more generality, we find an interesting paradox For instance, suppose the limits on the integral are from −A A to +A + A where A A is a real, positive number
- What is the difference between an indefinite integral and an . . .
Wolfram Mathworld says that an indefinite integral is "also called an antiderivative" This MIT page says, "The more common name for the antiderivative is the indefinite integral " One is free to define terms as you like, but it looks like at least some (and possibly most) credible sources define them to be exactly the same thing
- When does a line integral equal an ordinary integral?
One possible interpretation: a "normal" integral is simply a line integral where the path is straight and oriented along a particular axis Thus, as soon as you perform a transformation to the integrand to make the path straight and oriented properly, you're back at a "regular" integral
- What does it mean for an integral to be convergent?
The noun phrase "improper integral" written as $$ \int_a^\infty f (x) \, dx $$ is well defined If the appropriate limit exists, we attach the property "convergent" to that expression and use the same expression for the limit
- What is the integral of $e^{-x^2 2}$ over $\\mathbb{R}$
What is the integral of $$\int_ {-\infty}^ {\infty}e^ {-x^2 2}dx\,?$$ My working is here: = $-e^ (-1 2x^2) x$ from negative infinity to infinity What is the value of this?
- Integral of $\sqrt {1-x^2}$ using integration by parts
A different approach, building up from first principles, without using cos or sin to get the identity, $$\arcsin (x) = \int\frac1 {\sqrt {1-x^2}}dx$$ where the integrals is from 0 to z With the integration by parts given in previous answers, this gives the result The distance around a unit circle traveled from the y axis for a distance on the x axis = $\arcsin (x)$ $$\arcsin (x) = \int\frac
- Contour Integral Involving Cauchys Principal Value
I need help with the following integral $$ \mathrm {P}\!\int_ {-\infty}^ {\infty} \frac {\cos x} {x+a}\,dx, \qquad 0<a<1 $$ P is Cauchy's Principal Value The first
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