1) Introduction: Building on Antiderivatives 📚
In the last topic, we learned about antiderivatives and how to find them by "undoing" derivatives. Now, we’re going to take it further with indefinite integrals and introduce some basic rules to make the process smoother. These rules are like shortcuts that help us integrate a wide variety of functions without guessing every time!
In this topic, we’ll cover:
- Indefinite Integrals: A formal way to express the general antiderivative.
- Basic Integration Rules: Simple patterns to integrate polynomials, trigonometric functions, and more.
- Verification: How to check our work by differentiating.
Quick Recap: An antiderivative \( F(x) \) of \( f(x) \) satisfies \( F'(x) = f(x) \), and the general form is \( \int f(x) \, dx = F(x) + C \).
2) Indefinite Integrals: The General Antiderivative Revisited 🎓
The indefinite integral is just another name for the general antiderivative we explored before. It represents all possible functions whose derivative is the given function, plus a constant. This constant (\( C \)) accounts for the fact that derivatives ignore constant shifts.
Definition 9.1: Indefinite Integral
The indefinite integral of a function \( f(x) \) is written as:
\( \int f(x) \, dx = F(x) + C \)
where \( F'(x) = f(x) \), and \( C \) is the constant of integration. The \( dx \) indicates the variable of integration.
Think of the indefinite integral as a family of functions, all differing by a constant. For example, if \( f(x) = 2x \), then \( \int 2x \, dx = x^2 + C \), where \( C \) can be any number.
Example 1: Simple Indefinite Integral
Find \( \int 4x \, dx \).
We need \( F(x) \) such that \( F'(x) = 4x \).
- Try \( F(x) = 2x^2 \) (since \( \frac{d}{dx}(2x^2) = 4x \)).
- Add the constant: \( F(x) = 2x^2 + C \).
Answer: \( \int 4x \, dx = 2x^2 + C \).
Example 2: Constant Function
Find \( \int 7 \, dx \).
A constant’s derivative is zero, so we need a linear function.
- Try \( F(x) = 7x \) (since \( \frac{d}{dx}(7x) = 7 \)).
- Add the constant: \( F(x) = 7x + C \).
Answer: \( \int 7 \, dx = 7x + C \).
3) Basic Integration Rules 📐
To make integration easier, we can use some basic rules derived from differentiation. These rules help us integrate polynomials, constants, and simple trigonometric functions. Let’s explore them one by one!
Rule 9.1: Power Rule for Integration
For any real number \( n \neq -1 \):
\( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)
This is the reverse of the power rule for differentiation (\( \frac{d}{dx}(x^n) = n x^{n-1} \)).
Rule 9.2: Constant Multiple Rule
For a constant \( k \):
\( \int k f(x) \, dx = k \int f(x) \, dx \)
You can factor out constants when integrating.
Rule 9.3: Sum Rule
For functions \( f(x) \) and \( g(x) \):
\( \int [f(x) + g(x)] \, dx = \int f(x) \, dx + \int g(x) \, dx \)
Integrate each term separately.
Rule 9.4: Trigonometric Integrals
- \( \int \sin(x) \, dx = -\cos(x) + C \)
- \( \int \cos(x) \, dx = \sin(x) + C \)
These come from the derivatives of sine and cosine.
Example 3: Using the Power Rule
Find \( \int x^3 \, dx \).
Apply the power rule with \( n = 3 \):
- \( \int x^3 \, dx = \frac{x^{3+1}}{3+1} + C = \frac{x^4}{4} + C \).
Answer: \( \int x^3 \, dx = \frac{x^4}{4} + C \).
Example 4: Combining Rules
Find \( \int (2x^2 + 5x) \, dx \).
Use the sum rule and constant multiple rule:
- \( \int (2x^2 + 5x) \, dx = \int 2x^2 \, dx + \int 5x \, dx \).
- For \( 2x^2 \): \( \int 2x^2 \, dx = 2 \cdot \frac{x^{2+1}}{2+1} = 2 \cdot \frac{x^3}{3} = \frac{2x^3}{3} \).
- For \( 5x \): \( \int 5x \, dx = 5 \cdot \frac{x^{1+1}}{1+1} = 5 \cdot \frac{x^2}{2} = \frac{5x^2}{2} \).
- Combine: \( \frac{2x^3}{3} + \frac{5x^2}{2} + C \).
Answer: \( \int (2x^2 + 5x) \, dx = \frac{2x^3}{3} + \frac{5x^2}{2} + C \).
Example 5: Trigonometric Integral
Find \( \int (3\sin(x) - 2\cos(x)) \, dx \).
Use the sum rule and trigonometric rules:
- \( \int 3\sin(x) \, dx = 3 \cdot (-\cos(x)) = -3\cos(x) \).
- \( \int -2\cos(x) \, dx = -2 \cdot \sin(x) = -2\sin(x) \).
- Combine: \( -3\cos(x) - 2\sin(x) + C \).
Answer: \( \int (3\sin(x) - 2\cos(x)) \, dx = -3\cos(x) - 2\sin(x) + C \).
4) Verification: Checking with Differentiation 🔍
The best way to ensure our integrals are correct is to differentiate the result. If the derivative matches the original function, we’ve done it right! This step is like a double-check for our work.
Example 6: Verify an Integral
Verify that \( \int (x^2 + 3) \, dx = \frac{x^3}{3} + 3x + C \) is correct.
Differentiate the result:
- \( \frac{d}{dx} \left( \frac{x^3}{3} + 3x + C \right) = \frac{3x^2}{3} + 3 + 0 = x^2 + 3 \).
It matches \( x^2 + 3 \), so the integral is correct!
Example 7: Another Verification
Verify \( \int \cos(x) \, dx = \sin(x) + C \).
Differentiate:
- \( \frac{d}{dx} (\sin(x) + C) = \cos(x) + 0 = \cos(x) \).
It matches \( \cos(x) \), confirming the integral is correct.
5) Practice Questions 🎯
Fundamental Practice Questions 🌱
Instructions: Find the general antiderivative (indefinite integral) for each function. 📚
\( \int 6x \, dx \)
\( \int x^4 \, dx \)
\( \int 3x^2 \, dx \)
\( \int 5 \, dx \)
\( \int (4x - 1) \, dx \)
\( \int (x^3 + 2x) \, dx \)
\( \int 2\sin(x) \, dx \)
\( \int 3\cos(x) \, dx \)
\( \int (x^2 - 4x + 2) \, dx \)
\( \int (5x^3 - 2x^2 + x) \, dx \)
\( \int \sin(x) + \cos(x) \, dx \)
Challenging Practice Questions 🌟
Instructions: These require deeper understanding or interpretation. 🧠
Find \( \int (2x^3 - 3x^2 + x - 4) \, dx \) and verify your answer by differentiating.
Compute \( \int (3x^2 + 1)^2 \, dx \) using the substitution method (hint: expand first).
If \( F(x) = \int (x^2 + 2x) \, dx \), find \( F(1) - F(0) \) assuming \( C = 0 \). What does this value represent?
Determine \( \int (4\cos(x) - 5\sin(x)) \, dx \) and check your result.
Explain why \( \int x^{-2} \, dx \) requires special attention and find its indefinite integral.
6) Summary & Cheat Sheet 📋
6.1) Indefinite Integrals
\( \int f(x) \, dx = F(x) + C \), where \( F'(x) = f(x) \) and \( C \) is the constant of integration.
6.2) Basic Integration Rules
- Power Rule: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) (if \( n \neq -1 \))
- Constant Multiple: \( \int k f(x) \, dx = k \int f(x) \, dx \)
- Sum Rule: \( \int [f(x) + g(x)] \, dx = \int f(x) \, dx + \int g(x) \, dx \)
- Trigonometric: \( \int \sin(x) \, dx = -\cos(x) + C \), \( \int \cos(x) \, dx = \sin(x) + C \)
6.3) Verification Tip
Always differentiate your answer to confirm it matches the integrand.
You’re now equipped with the basic tools of integration! Next, we’ll explore more advanced techniques like substitution. 🎉