# Indefinite Integration problem

• QuarkCharmer
In summary, to find the position of a particle moving with the given data, you integrate the acceleration function to get the velocity function, and then integrate the velocity function to get the position function. Using the parameters s(0)=0 and s(1)=20, you can solve for the constants and determine the position of the particle at any given time.
QuarkCharmer

## Homework Statement

A particle is moving with the given data. Find the position of the particle.
$a(t)=t^{2}-4t+6$,
$s(0)=0$,
$s(1)=20$

## The Attempt at a Solution

$a(t)=t^{2}-4t+6$, $s(0)=0$, $s(1)=20$
$$v(t)=\int t^{2}-4t+6 dt$$
$$v(t)=\frac{t^{3}}{3}-2t^{2}+6t+C_{1}$$
Then I suppose I take the antiderivative again to get to s (distance), I can't solve for the constant yet.

$$s(t)=\int \frac{t^{3}}{3}-2t^{2}+6t+C_{1} dt$$
$$s(t)=\frac{t^{4}}{12}-\frac{2t^{3}}{3}+3t^{2}+C_{1}t+C_{2}$$

So now I have two constants in there? Would I just use the parameters $s(0)=0$ and $s(1)=20$ to try to solve this like a system of 2 equations with two unknowns? I don't really know how to proceed. The other problems like this at least gave me a value for the first derivative of the original function so I could sort of work backwards.

Last edited:
QuarkCharmer said:
So now I have two constants in there? Would I just use the parameters $s(0)=0$ and $s(1)=20$ to try to solve this like a system of 2 equations with two unknowns?

This is exactly what you do. Be more confident!

## What is indefinite integration?

Indefinite integration is a mathematical process of finding the most general antiderivative of a given function. It involves reversing the process of differentiation to find an expression for the original function.

## What is the difference between indefinite and definite integration?

The main difference between indefinite and definite integration is that indefinite integration gives a general solution in the form of a constant, while definite integration gives a specific numerical value for the integral over a given interval.

## What are the basic rules of indefinite integration?

The basic rules of indefinite integration include the power rule, constant multiple rule, sum and difference rule, and the substitution rule. These rules help to simplify the process of finding antiderivatives of functions.

## How do you solve an indefinite integration problem?

To solve an indefinite integration problem, you first need to identify the function and apply the appropriate integration rule. Then, you integrate the function and add a constant of integration at the end.

## What are some common techniques for solving indefinite integration problems?

Some common techniques for solving indefinite integration problems include integration by parts, trigonometric substitution, and partial fractions. These techniques can be used to solve more complex integration problems that cannot be solved using the basic rules.

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