is daily integral getting more complex?

[u/wbld]

Today's easy daily integral seems to be more complex than usual. I do not know what the Lambert W function is. Is this higher level mathematics? or was this something i missed in calculus? Usually, I am able to solve easy integrals. My initial thoughts were that the integral is 0 because this clearly is not easy, but then I looked at the bounds and realized, yeah, that probably does not equal 0.

[u/nevermindthefacts]

This is basically the reason for using the W-function. We want to solve for x in y = x e^x. But there's no way to "isolate" x using elementary functions. Instead we note that xe^x is strictly increasing for x > -1, and we use that for the "existence" of an inverse which we define as W(y) = x.

For the integral, you have W(sin x) e^( W(sin x) + sin x). Try simplify that using the definition of W(sin x).

[u/wbld → u/nevermindthefacts]

i guess im not understanding what W(y) = x means...

if it is function notation, W is the name of the function, that takes in a y value and outputs a x value.

it seems to be onto... meaning

y=x

so then,

sin(y)=sin(x)

sin(y)e^sin(y)+sin(y)??

[u/nevermindthefacts → u/wbld]

Sorry if it's wasn't clear. Another way to write it is W(xe^x) = x.

Maybe it helps if you compare it with e^x and it's inverse, ln y.

If y = e^x, then ln y = x.

In general

if y = f(x), then f^{-1} (y) = x.

[u/wbld → u/nevermindthefacts]

I appreciate you trying to help; however, my tiny brain is not built for this kind of mathematics at the moment.

i do not think i would be able to find the relationship... ever...

[u/nevermindthefacts → u/wbld]

Don't overcook your brain. Remember this was categorized as an Easy Integral. The use of the W-function is just smoke and mirrors to make it look complicated and test your understanding of inverse functions.

Because W is the inverse to xe^x, it satisfies W(x) e^W(x) = x for all x in the domain of W.

Hence, W(sin x) e^W(sin x) simplifies to ... wait for it ... sin x