# Find Velocity of Particles: Indefinite Integrals

• MHB
In summary, to find the velocity of particles, one must evaluate the indefinite integral of the acceleration function, a(t), using the formula v = Z a(t) dt. IBP can be used to obtain the formula \int 125t^4\ln^2(t)\,dt = t^5\left(25\ln^2(t)-10\ln(t)+2\right)+C. Further explanation and understanding of IBP can be found here.
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To help find the velocity of particles requires the evaluation of the indefinite integral of the acceleration
function, a(t), i.e.
v = Z a(t) dt.

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I would think IBP could be used to obtain:

$$\displaystyle \int 125t^4\ln^2(t)\,dt=t^5\left(25\ln^2(t)-10\ln(t)+2\right)+C$$

If you want help actually deriving this formula, please let me know. :)

For some useful information in understanding IBP, see here.

\displaystyle \begin{align*}\int 125t^4\ln^2(t)\,dt&=25t^5\ln^2(t)-50\int t^4\ln(t)\,dt \\ &=25t^5\ln^2(t)-50\left(\frac15t^5\ln(t)-\frac15\int t^4\,dt\right) \\ &=25t^5\ln^2(t)-10t^5\ln(t)+2t^5+C\end{align*}

## 1. What is a particle velocity?

A particle velocity is the rate of change of position of a particle with respect to time. It is a vector quantity, meaning it has both magnitude and direction.

## 2. How is the velocity of particles calculated?

The velocity of particles can be calculated by taking the derivative of the position function with respect to time. This can be done using calculus, specifically using the derivative of the position function known as the instantaneous rate of change, or by taking the slope of the position-time graph.

## 3. What is an indefinite integral?

An indefinite integral is an operation that, given a function, finds another function whose derivative is equal to the original function. It is represented by the symbol ∫ and is the inverse operation of differentiation.

## 4. How are indefinite integrals used to find velocity of particles?

Indefinite integrals are used to find the velocity of particles by integrating the acceleration function with respect to time. This yields the velocity function, which can then be evaluated at specific time points to find the velocity of the particle at those times.

## 5. What are some real-world applications of finding velocity of particles using indefinite integrals?

Finding velocity of particles using indefinite integrals has many practical applications, such as in physics, engineering, and even daily life. For example, it can be used to calculate the speed of a moving object, the rate of change of temperature in a system, or the growth rate of a population over time.

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