These identities for $\int_0^1 x^ {-x}\ dx$ and $\int_0^1 x^x\ dx$ are sometimes called the "sophomore's dream". Look that up on Wikipedia.

The symbol used for integration, $\int$, is in fact just a stylized "S" for "sum"; The classical definition of the definite integral is $\int_a^b f (x) dx = \lim_ {\Delta x \to 0} \sum_ {x=a}^ {b} f (x)\Delta x$; the limit …

May 2, 2015 · Okay this may sound stupid but I need a little help... What do $\Large \frac {d} {dx}$ and $\Large \frac {dy} {dx}$ mean? I need a thorough explanation. Thanks.

Sep 9, 2020 · But then others told me that "dx" is part of what's being integrated, and they started saying that we're led to believe that its just a delimiter in early courses because it'd be impossible for …

Dec 27, 2013 · Well, $\delta x$ means different things depending on the context. For example, it has a particular meaning in variational calculus, and a completely different one in functional calculus...

Jan 9, 2015 · I'm taking differential equations right now, and the lack of fundamental knowledge in calculus is kicking my butt. In class, my professor has done several implicit differentiations. I realize …

A "signed definite integral" for computing work and other "net change" calculations. The value of an expression such as $\int_0^1 x^2\,dx$ comes out the same under all these interpretations, of course. …

Here, $ (dx)^2$ means $dx \wedge dx$, and the fact that it vanishes comes from the fact that the exterior algebra is anti-commutative. In other words, formally we have $d^2x=0$ and $ (dx)^2=0$ but …

I am working on trying to solve this problem: Prove: $\\int \\sin^n{x} \\ dx = -\\frac{1}{n} \\cos{x} \\cdot \\sin^{n - 1}{x} + \\frac{n - 1}{n} \\int \\sin^{n - 2}{x ...

Mar 5, 2011 · In Leibniz notation, the 2nd derivative is written as $$\dfrac {\mathrm d^2y} {\mathrm dx^2}\ ?$$ Why is the location of the $2$ in different places in the $\mathrm dy/\mathrm dx$ terms?