Question

For the function f(t) = (cos√t )/(√t) , find the antiderivative F of f that satisfies the given condition F(pi^2 / 4) = 5.

Answer 1

You could perform a substitution

Answer 2

I'm confused...Why did you post a link to this page?

Answer 3

\[\int\limits_{}^{}\frac{\cos(\sqrt{t})}{\sqrt{t}} dt \]
do u=sqrt(t)

Answer 4

never mind, can u just help me?

Answer 5

yes its cosine of sqrt of t

Answer 6

let me know if you need more help after trying that sub

Answer 7

can you pls help I have no clue how to do this. please?

Answer 8

@freckles

Answer 9

\[u=\sqrt{t} \\ \frac{du}{dt}=?\]

Answer 10

do you know how to find the derivative of sqrt(t)?

Answer 11

and remember that is just t^(1/2)

Answer 12

use power rule

Answer 13

derivative would be 1/(2√t )right?

Answer 14

yes

Answer 15

\[u=\sqrt{t} \\ 2 du =\frac{1}{\sqrt{t}} dt \]
tried to use this substitution in your integral

Answer 16

okay! I get that part now that you showed, but then what?

Answer 17

i know you have to take the integral but idk integral of what

Answer 18

\[F(t)=\int\limits \frac{\cos(\sqrt{t})}{\sqrt{t}} dt \]
replace the sqrt(t) inside the cosine with u
replace the 1/sqrt(t) dt with 2 du

I'm asking you to use the sub

Answer 19

\[F(t) =\int\limits \cos(\sqrt{t}) \frac{1}{\sqrt{t}} dt \]
\[F(t)=\int\limits \cos(u) 2 du =2 \int\limits \cos(u) du\]
Can you integrate cos(u) w.r.t. u ?

Answer 20

no.. can u just help me all the way through this problm? i have a hw page full of these type of questions I promise I'll do them all!

Answer 21

so you don't know what function you can take derivative of that will give you cos?

Answer 22

\[( ? )'=\cos(u)\]

Answer 23

sine

Answer 24

yes

Answer 25

sorry i didnt know what u meant

Answer 26

so F(t)=2sin(u)+C
where u=sqrt(t)

Answer 27

ok and then?

Answer 28

so thats the antiderivative right?

Answer 29

but idk how to satisfy the other equation!

Answer 30

Well you have an unknown constant C there

Answer 31

use the condition to find it

Answer 32

F(a)=b
means you will replace t with a
and F(a) with b

Answer 33

can you show me?

Answer 34

let me see you try to plug in the values

Answer 35

replace t with pi^2/4 and F(pi^2/4) with 5
and solve for C

Answer 36

can you show m actually like you did above w/ pictures? not just words becuz its hard for me to uderstand

Answer 37

You aren't willing to try to apply what I said?

Answer 38

im sorry i just... its not that im not willing i have to see it to understand it.

Answer 39

Do you know what we found for F(t)?

Answer 40

so 2sin√(pi^2/4) +c = what?

Answer 41

can you tell me the function so far?

Answer 42

well don't forget what that expression equals
\[F(t)= 2 \sin(\sqrt{t})+C\]

Answer 43

so then?

Answer 44

you are going to replace all the t's with pi^2/4\[F(\frac{\pi^2}{4})=2 \sin(\sqrt{\frac{\pi^2}{4}})+C\]

Answer 45

And you are given what F(pi^2/4) equals

Answer 46

so replace F(pi^2/4) with 5 since F(pi^2/4)=5