Question

A particle starts at x(0) = 2. If its velocity is given by v(t) = ln(1+t), find its position at t = 5

Answer 1

the primitive of v(t) will give the displacement equation x(t)

so find the indefinite integral of

\[\int\limits \ln(1 + t) dt\]

you also know that when t = 0, x(t) = 2 so the particle starts 2 units to the right of the origin.

you will need to use these to evaluate the constant in the indefinite integral.

Lastly,

when you have your equation, substitute t = 5 and evaluate to find the postiion from the origin

Answer 2

by u substitution correct? i think i am doing that process wrong

Answer 3

well its a standard integral

\[\int\limits \ln(ax + b) dx = \frac{(ax + b)}{a} \ln(ax + b) - x + c\]

Answer 4

oh that is where i get confused. lol. because i did:

y = lnu

u = 1+t

dy/du = ulnu

du/dx = 1

dy/dx = dy/du * du/dx

and then i add that to the constant 2.

Answer 5

@zepdrix can you help me? loll maybe i'm thinking waaayy too much into this

Answer 6

you need to use integration by parts

so f = ln(u) dg = du
df = 1/u du g = u

so then you have

\[uln(u) - \int\limits 1 du\]

which gives the integral of

uln(u) - u

and when u = t + 1 you get

- t + (t + 1)(ln (t + 1) - 1)

or

(t + 1)log(t +1) - 1

Answer 7

AHHH ok thanks :)