# Integration of polynomial quotient and partial fractions

A example is shown whereby the velocity of an object is expressed as a quotient of two polynomials in time t.

Such that velocity v = (t^2 + 18t + 21)/{(2t+1)*(t+4)^2}

The problem asks to determine the displacement of the object after time t = 2.1s.

The process requires the integration of the polynomial between the limits. The integration of the polynomial is not possible on its own so a process whereby the polynomial expression for velocity is first broken down into its partial fractions which are then in turn integrated is shown.

The solution shows a step by step approach to show how the original expression(t^2 + 18t + 21)/{(2t+1)*(t+4)^2} is broken down to its constituent partial fractions of 1/(2t+1) + 5/(t+4)^2

The individual partial fractions are then integrated and the limits for the integration put in to determine the displacement over this time period.

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#### Solution Preview

Determining the displacement over time by using the integration of a polynomial for time (t) that defines the velocity at time t. A solution that involves algebraic manipulation to simplify the polynomial quotient into partial fractions and then carrying out the integration.

Under certain conditions, the velocity v (in m/s) of an object moving along a straight line as a function of the time t (in s) is given by

v= (t^2 +18t+21) / ((2t+1)*((t+4)^2))

Find the distance travelled by the object during the first 2.10 s.

We need to integrate the above function of t with respect to t but on its own this integration is difficult to perform unless we simplify the function into its partial fractions.

? Let us find the partial fractions so

(t^2 + 18t ...

#### Solution Summary

A example is shown whereby the velocity of an object is expressed as a quotient of two polynomials in time t.