Question

Integration question: integral cos(lnx)

Answer 1

Hmm this is one of those annoying ones where you have to do integration by parts, then do some algebra, combining integrals to solve for the thing...

Ok lemme see if we can get this setup correctly :)

Answer 2

\[\huge \int\limits \cos(\ln x) dx\]
\[\large u=\color{salmon}{\cos(\ln x)}, \quad dv=\color{blueviolet}{dx}\]\[\large du=\color{orangered}{-\frac{1}{x}\sin(\ln x) dx}, \quad v=\color{purple}{x}\]That part make sense? :o

Answer 3

yeah

Answer 4

\[\large \color{purple}{x}\color{salmon}{\cos(\ln x)}+\int\limits \frac{\color{purple}{x}}{\color{orangered}{x}}\color{orangered}{\sin(\ln x) dx}\]So we get something like this i think.

Answer 5

yeah

Answer 6

We'll have to do Integration by parts again.
It's important that you assign the similar pieces for your by parts.

Our trig was our U, so it'll have to be our U again.

Answer 7

\[\large u=\color{\tan}{\sin(\ln x)}, \quad dv=\color{limegreen}{dx}\]\[\large du=\color{goldenrod}{\frac{1}{x}\cos(\ln x)dx},\quad v=\color{forestgreen}{x}\]

Answer 8

Sorry If I'm moving way too slow, I'm trying to experiment with the colors :) lol

Answer 9

Are you sure you can do integration by parts here?

Answer 10

Makes it much more interesting :)

Answer 11

Woah, I see where you're going with this

Answer 12

Which gives us,\[\large \color{purple}{x}\color{salmon}{\cos(\ln x)}+\color{forestgreen}{x}\color{\tan}{\frac{1}{x}\sin(\ln x)}-\int\limits \frac{\color{forestgreen}{x}}{\color{goldenrod}{x}}\color{goldenrod}{\cos(\ln x)dx}\]Check my work make sure I have those plugged in right hera :O

Answer 13

yeah that's right

Answer 14

Hmm notice how we got back what we started with? It might make you think we took a wrong turn, but this is good. We can do something sneaky from here.

Answer 15

Let,\[\large Y=\int\limits \cos(\ln x) dx\]It will help us save time with writing.
So what we have right now is,\[\large Y=x \cos(\ln x)+x \sin(\ln x)-Y\]

Answer 16

See where this is going? :D

Answer 17

That's brilliant :D

Answer 18

Can you finish it from here? :3

Answer 19

yeah y=(xcos(lnx)+xsin(ln(x)))/2

Answer 20

Yay team \:D/

Answer 21

Hehe, I remember doing this exact problem in Calc 2. That's funny :D

Answer 22

lol that's so cool that it works like that.