calculus - Integration by parts on definite integral - Mathematics ...
Feb 28, 2026 · I have an integral, $$ I = \int_a^b x f (x) dx $$ and I would like to express this in terms of $\int_a^b f (x) dx$ if possible, but I don't see how integration by parts will help here.
Why must the curve of an integral intersect the origin?
Jan 4, 2026 · The other kind of integral you often encounter is the definite integral. This is the integral that is sometimes described as "the area under the curve" (although I would consider that an …
Various methods for Integral from MIT Integration Bee 2026 Semifinal
Mar 23, 2026 · Encountering the integral $$ \int \frac {x^2-2} {\left (x^2+2\right) \sqrt {x^4+4}} d x, $$ from MIT integration 2026 Semifinal , I tried my best to finish it within the time limit. $$ \begin {aligned} ...
What is the integral of 0? - Mathematics Stack Exchange
Feb 4, 2018 · The integral of 0 is C, because the derivative of C is zero. Also, it makes sense logically if you recall the fact that the derivative of the function is the function's slope, because any function f …
What is the difference between an indefinite integral and an ...
Nov 29, 2013 · 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 …
calculus - Why is the area under a curve the integral? - Mathematics ...
Dec 23, 2010 · One is the question of why the definite Riemann integral gives the correct notion of "area under a curve" for a (nonnegative, Riemann integrable) function. The other, which seems to be what …
What other tricks and techniques can I use in integration?
Apr 12, 2021 · So far, I know and can use a reasonable number of 'tricks' or techniques when I solve integrals. Below are the tricks/techniques that I know for indefinite and definite integrals separately. …
What is an integral? - Mathematics Stack Exchange
Dec 15, 2017 · 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 …
solving the integral of $e^ {x^2}$ - Mathematics Stack Exchange
The integral which you describe has no closed form which is to say that it cannot be expressed in elementary functions. For example, you can express $\int x^2 \mathrm {d}x$ in elementary functions …
Can the integral closure of a ring be taken intrinsically?
Oct 11, 2025 · However, one "intrinsic integral closure" that is often used is the normalization, which in the case on an integral domain is the integral closure in its field of fractions. It's the maximal integral …