Question

indefinite integral cos(t^1/2)/(t^1/2)

Answer 1

\[\int\limits_{}^{}\frac{ \cos \sqrt{t} }{ \sqrt{t} }\]

Answer 2

try \(u=\sqrt{t}\)

Answer 3

@satellite73 always beats me to the punch lol :D

Answer 4

i dont understand how u subtitution really works. can someone explain what do i do after finding u?

Answer 5

@satellite73 @jagr2713

Answer 6

ok lets say you have

5(2x+3)=3(4x+8)

you just multiply 5x2 and 5x3 and 3x4 and 3x8

so you will get

10x+15=12x+24

then you just solve for x

Answer 7

i still dont understand. once i find out u = sqrt(t) then what do i do?

Answer 8

@jagr2713

Answer 9

oh you mean substitution of your problem. i dont know how to do that sorry

Answer 10

rewrite the integral in terms of u and du. So what is du?

Answer 11

that is give me
\[D[\sqrt{t}]\]

Answer 12

du=1/2sqrt(t) @sylbot

Answer 13

so?? just plug it in
u = \(\sqrt t\rightarrow du=\dfrac{1}{2\sqrt t}dt\)
your integral turns to
\[\int 2cosu du= 2\int cos u du\]
It's quite easy now, right?

Answer 14

the derivative of \[\sin(\sqrt{t})\] is
\[\cos(\sqrt{t})\times \frac{1}{2\sqrt{t}}\] by the chain rule
that would be your answer, except you are off by a factor of two

Answer 15

Looks like you've got plenty of quality help already, I just wanted throw in my two cents.

Personally I like to rewrite the integral, separate things a little bit,
so it will be easier to see where everything is getting plugged in.\[\Large\rm \int\limits \frac{\cos \sqrt t}{\sqrt t}dt=\int\limits \cos\color{royalblue}{\sqrt t}\left(\color{orangered}{\frac{1}{\sqrt t}dt}\right)\]Then you're making the substitution:\[\Large\rm \color{royalblue}{u=\sqrt t}\]Taking your derivative gives,\[\Large\rm du=\frac{1}{2\sqrt t}dt\qquad\to\qquad \color{orangered}{2du=\frac{1}{\sqrt t}dt}\]So then plugging in our pieces,\[\Large\rm \int\limits\limits \cos\color{royalblue}{\sqrt t}\left(\color{orangered}{\frac{1}{\sqrt t}dt}\right)=\int\limits\limits \cos\color{royalblue}{u}\left(\color{orangered}{2du}\right)\]

Answer 16

Thank you actually the last one helped me out the most understanding!