How Do You Undo A Rate with the Substitution Method?

Suppose that the profit for a company is increasing at a rate of


where the company has been in operation for t years. What is the total change in profit over the first three years?

In this problem, we are given the rate at which profit is changing over time. This is confirmed by the fact that the function is defined as P′(t), the derivative of profit. However, the question is about the corresponding profit function P(t). So we need to find this profit function by taking the antiderivative of P′(t),


Remember, the antiderivative undoes the derivative so the antiderivative of P′(t) is P(t). To do this antiderivative, we need to use the Substitution Method.


This means that


You might think that the total change in profit over the first three years is P(3), but this is the profit at the end of the third year. To find the total change in profit we need to calculate P(3) – P(0),



How Do You Use the Substitution Method to Calculate Change?

Only one problem on the Section 14.1 Homework was missed by very many people. In that problem you were given the rate of change of profit, P‘(t), and asked to calculate how much the profit changed. Since this is a question about P(t), you need to undo the derivative with an antiderivative in the form of the Fundamental Theorem of Calculus. With this function, we would write it as

$latex displaystyle intlimits_{a}^{b}{P'(t),dx=P(b)-P(a)}$

Continue reading “How Do You Use the Substitution Method to Calculate Change?”

How Do You Find an Antiderivative Using the Substitution Method?

In Monday’s class, student found many antiderivatives using the Substitution Method. The basic process is illustrated below for the antiderivative

$latex \displaystyle \int{4{{\left( {{x}^{2}}-3 \right)}^{3}}\cdot 2x,dx}$.

Let’s look at the basic steps.

  1. Choose the expression for u. This is generally the inside part of a composition in the integrand. Use the derivative to find an other expression for du.
  2. Match the integrand with u and du. All variables in the original integrand must change to u.
  3. Change the integrand so that it is written in terms of u.
  4. Work out the antiderivative in terms of u.
  5. Put in the expression for u so that the antiderivative is written in terms of the original variable.

Now let’s look at the examples carried out in class.

Problem 1

$latex \displaystyle \int{{{\left( 3{{x}^{2}}+4 \right)}^{4}}\cdot 6x,dx}$

Problem 2

$latex \displaystyle \int{{{\left( 3{{x}^{2}}+4 \right)}^{3}}\cdot 4x,dx}$

Problem 3

$latex \displaystyle \int{{{\left( {{x}^{2}}-1 \right)}^{5}}\cdot x,dx}$

Problem 4

$latex \displaystyle \int{4{{\left( 2x+3 \right)}^{2}},dx}$

Problem 5

$latex \displaystyle \int \frac{2}{{{\left( 2m+1 \right)}^{3}}} dm$


Problem 6

$latex \displaystyle \int{\frac{3}{\sqrt{3u-5}},du}$

Problem 7

$latex \displaystyle \int{-4{{e}^{2p}},dp}$

Problem 8

$latex \displaystyle \int{5{{e}^{-0.3g}},dg}$