A particle is moving along a line so that if v feet per second is the velocity of the particle at t second, then v= t^2-t+1/((t+2)^2(t^2+1)). Find a formula for the distance traveled by the particle from the time when ๐‘ก = 0 ๐‘ก๐‘œ ๐‘กโ„Ž๐‘’ ๐‘ก๐‘–๐‘š๐‘’ ๐‘คโ„Ž๐‘’๐‘› ๐‘ก = t sub 1

that would, of course, be

โˆซ[0,t_1] v(t) dt
You'll have to use partial fractions to do this one.

To find the formula for the distance traveled by the particle from ๐‘ก = 0 to ๐‘ก = ๐‘กโ‚, we need to integrate the velocity function with respect to time over that interval.

Given that the velocity function v(t) = tยฒ - t + 1 / ((t + 2)ยฒ(tยฒ + 1)), we'll find the antiderivative of v(t) and then evaluate it at ๐‘ก = ๐‘กโ‚ and ๐‘ก = 0.

Step 1: Find the antiderivative of v(t)
To find the antiderivative, we integrate each term separately:

โˆซ(tยฒ - t + 1) / ((t + 2)ยฒ(tยฒ + 1)) dt

Let's break it down:

โˆซ(tยฒ - t + 1) / ((t + 2)ยฒ(tยฒ + 1)) dt
= โˆซtยฒ / ((t + 2)ยฒ(tยฒ + 1)) dt - โˆซt / ((t + 2)ยฒ(tยฒ + 1)) dt + โˆซ1 / ((t + 2)ยฒ(tยฒ + 1)) dt

Step 2: Evaluate the antiderivative at ๐‘ก = ๐‘กโ‚ and ๐‘ก = 0
Let F(t) be the antiderivative of v(t):
F(t) = โˆซtยฒ / ((t + 2)ยฒ(tยฒ + 1)) dt - โˆซt / ((t + 2)ยฒ(tยฒ + 1)) dt + โˆซ1 / ((t + 2)ยฒ(tยฒ + 1)) dt

Now, evaluate F(t) at ๐‘ก = ๐‘กโ‚ and ๐‘ก = 0:

Distance traveled = F(๐‘กโ‚) - F(0)

To find the specific formula for the distance traveled, we'll need to calculate each of the integrals separately, evaluate them at ๐‘ก = ๐‘กโ‚ and ๐‘ก = 0, and then subtract F(0) from F(๐‘กโ‚).

Unfortunately, finding an explicit formula for the distance traveled may not be feasible without specific values for ๐‘กโ‚. However, you can calculate the distance traveled by evaluating the definite integral numerically using numerical methods or software such as a graphing calculator or computer algebra system (CAS).