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What is an integral? - Mathematics Stack Exchange A different type of integral, if you want to call it an integral, is a "path integral" These are actually defined by a "normal" integral (such as a Riemann integral), but path integrals do not seek to find the area under a curve I think of them as finding a weighted, total displacement along a curve
What is the integral of 1 x? - Mathematics Stack Exchange Answers to the question of the integral of $\frac {1} {x}$ are all based on an implicit assumption that the upper and lower limits of the integral are both positive real numbers
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
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
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