Distance Traveled by Point in Time Interval [1,3]

[sarmen]

Homework Statement

suppose the velocity of a point moving at time t, in seconds along a coordinate line is v(t)= (t+3)/(t^3+t) ft/sec. how far does the point travel during the time interval [1,3].

Homework Equations

The Attempt at a Solution

im not sure what to do, i used ∫3 (t+3)/(t^3+t) dt
this goes under the integral>> 1

but then i don't know what do next. please help

 

[Mark44]

Homework Statement

suppose the velocity of a point moving at time t, in seconds along a coordinate line is v(t)= (t+3)/(t^3+t) ft/sec. how far does the point travel during the time interval [1,3].

Homework Equations

The Attempt at a Solution

im not sure what to do, i used ∫3 (t+3)/(t^3+t) dt
this goes under the integral>> 1

but then i don't know what do next. please help

Rewrite (t + 3)/(t³ + t) using partial fractions decomposition. The denominator factors into t(t² + 1), and the decomposition should look like A/t + (Bt + C)/(t² + 1), for some constants A, B, and C.

Decomposition will give you two integrals to evaluate.

 

[Quinzio]

You can try splitting the denominator into its components.

 

[sarmen]

ok so i did the partial fraction decomposition and got (3/t)+ (-3t+1)/ (t² + 1).

do i dt it or just use the integrals [1,3] and plug it in?

 

[sarmen]

anyone?

 

[olivermsun]

If
$$f(t) = g(t) + h(t)$$
then
$$\int_a^b f(t) dt = ?$$

 

[sarmen]

If
$$f(t) = g(t) + h(t)$$
then
$$\int_a^b f(t) dt = ?$$

dont we use the partial fraction that i got?

 

[Mark44]

Yes. olivermsun is telling you that you can rewrite an integral that is a sum of functions as the sum of the integrals of the functions. For your problem, you probably want to split it into three integrals.

 

[sarmen]

if i got (3/t)+ (-3t+1)/ (t2 + 1) then it would be

∫3/t dt + ∫(-3t/(t^2 + 1))dt +∫1/(t^2 + 1)dt

then I am not so sure
3ln|t|+ ? + tan^-1(x/1).

i don't know how to do ∫(-3t/(t^2 + 1))dt

 

[Norfonz]

Consider the change of variables u = t² + 1.

 

[Mark44]

Consider the change of variables u = t² + 1.

IOW, an ordinary substitution.