Question

I need help solving the following:

ds/dt = 6sin(t + pi), s=0 @ t=0.

I rewrote it like this:
ds/dt = 6sin(t + pi)dt

I know that one has to take the anti-derivative, plug in the values (0) for "s" & "t" to determine the value of C, and then write as equal to s (in Slope-Intercept Form), but before most of all of that...
I intend(ed) to solve this using u-substitution, as I have several like it, but what do I use as "u" that will allow me to write the original derivative/equation in terms of u?

*My initial instinct is to use "sin(t + pi)". My second guess is just "(t + pi)".

Any and all help is greatly appreciated! :)

Answer 1

Can I simply take the anti-derivative of the original... (what is it even called)... "Derivative"?

Answer 2

Any advice ? :/

Answer 3

Derivative of

sin(x + c) where c is a constant, is cos(x+pi)

Answer 4

Yep, I got that:) so, would 6dt (the rest of the original... thing..., aside from u) then be equal to 1/6(cos(t + pi ))du ?

Answer 5

No need for u sub.

Integral of

sin(x+pi) is -cos(x+pi)

Answer 6

I can write it as an integral?? :D

Answer 7

Well you're finding the antiderivative

Answer 8

Right... So, is an indefinite integral though, right?
So, it's the integral of:
6(sin(t + pi))dt

Answer 9

... But then, if there's no substitution, does it anti-derive as:
6(-cos(t + pi))(t^2 + pi(t^(0 + 1))) = 6(-cos(t + pi ))(t^2 + t) ??

Answer 10

Integral of

sin(x+pi)

is

-cos(x+pi)

Answer 11

The 6 makes no difference

Answer 12

.... So that coefficient to the trig function just goes away? Amd what about the quantity of theta? Doesn't that get "anti-derived", chain rule style?

Answer 13

Integral of

asin(x+pi)

is

-acos(x+pi) + C

if a is a constant

Answer 14

Okay, that makes sense. But touching the quantity of theta was wrong?

Answer 15

You're making it way more complicated than it is :P

Use differentiation to help you.

What's the derivative of 5sin(x + 3) ?

Answer 16

5cos(x + 3) (?)

Answer 17

Yes. You can use the chain rule, but the derivative of x+3 is just 1 anyway.