Table of Contents

1. Introduction
2. Finding the Constant of Integration
   • Key Concepts
   • Steps to Find the Constant
3. Worked Example
4. Practice Questions
5. Glossary
6. Summary and Key Takeaways
7. Exam Strategy

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Introduction

In calculus, finding the constant of integration is a crucial step when solving indefinite integrals. This process allows us to determine the specific form of an antiderivative that fits given conditions.

Finding the Constant of Integration

Key Concepts

Steps to Find the Constant

1. Integrate the Function:
   • Perform the integration to find the general form of the antiderivative.
2. Use Additional Information:
   • If additional information is provided, such as a specific value of $F(x)$ for a given $x$, or a point $(x_0, y_0)$ that the graph passes through, use this to find $C$.
3. Set Up and Solve an Equation:
   • Use the given point to set up an equation and solve for $C$.

Example Process

Suppose we need to integrate $f(x) = 2x + 3$ and find the constant of integration given that $F(x)$ goes through the point $(1, 2)$.

1. Integrate: $$F(x) = \int (2x + 3) , dx = x^2 + 3x + C$$

2. Use Given Point:

   • Given $F(1) = 2$, substitute $x = 1$ and $F(x) = 2$ into the equation: $$2 = 1^2 + 3(1) + C$$

3. Solve for $C$: $$2 = 1 + 3 + C \Rightarrow 2 = 4 + C \Rightarrow C = -2$$

Thus, the specific antiderivative is: $$F(x) = x^2 + 3x - 2$$

Worked Example

The function $h$ measures the height of a projectile above the ground at time $t$ (in seconds). It is known that $h$ satisfies the equation:

$$h(t) = \int (70 - 32t) , dt$$

and at time $t = 2$, the projectile is 81 feet above the ground.

Find an explicit expression for $h$ in terms of $t$.

Solution

1. Integrate: $$h(t) = \int (70 - 32t) , dt = 70t - 16t^2 + C$$

2. Use Given Point:

   • Given $h(2) = 81$, substitute $t = 2$ and $h(t) = 81$ into the equation: $$81 = 70(2) - 16(2)^2 + C$$

3. Solve for $C$: $$81 = 140 - 64 + C \Rightarrow 81 = 76 + C \Rightarrow C = 5$$

Thus, the explicit expression for $h$ is: $$h(t) = 70t - 16t^2 + 5$$

Practice Questions

Practice Question

1. Find the constant of integration for the integral $\int (4x - 7) , dx$ given that the antiderivative passes through the point $(2, 1)$.

Practice Question

2. Integrate the function $f(x) = 3x^2 - 2x + 1$ and determine the constant of integration if the function passes through the point $(1, 4)$.

Glossary

• Constant of Integration ($C$): An arbitrary constant added to the antiderivative of a function when computing an indefinite integral.
• Indefinite Integral: An integral without upper and lower limits, representing a family of functions.
• Antiderivative: A function whose derivative is the given function.

Summary and Key Takeaways

• When finding an indefinite integral, always include the constant of integration, $C$.
• Use additional information, such as specific points or values, to solve for $C$.
• The process of finding $C$ is equivalent to finding the particular solution to a differential equation.

Exam Strategy

• Read the Problem Carefully: Ensure you understand all given conditions and points.
• Double-Check Integration: Make sure your integration is correct and includes $C$.
• Use Given Points Accurately: Substitute them correctly into the integrated function to find $C$.

By following these strategies and understanding the process, you'll be better prepared to handle questions involving constants of integration in your exams.