WhizMath

Introduction: Reversing the Flow of Change

Welcome to Whizmath, where we continue our journey through the powerful realm of calculus! After exploring derivatives, which measure instantaneous rates of change, we now turn our attention to Integrals. Integrals are essentially the inverse operation of derivatives, allowing us to reverse the process of differentiation and, more profoundly, to calculate accumulation, areas, and volumes.

While derivatives tell us how fast something is changing at a moment, integrals tell us the total amount of change that has occurred over an interval. This makes them indispensable in fields ranging from physics (calculating work done, displacement from velocity), to engineering (stress and strain analysis), economics (total cost, revenue), and statistics (probability distributions).

In this comprehensive lesson, we will define antiderivatives, distinguish between indefinite and definite integrals, introduce the fundamental theorem that links differentiation and integration, explore various integration techniques, and apply integrals to solve real-world problems. Prepare to unlock the power of accumulation and gain a deeper understanding of quantities built up over time!

Chapter 1: The Concept of an Integral (Antiderivatives)

Integration begins with the idea of finding a function whose derivative is a given function.

1.1 Antiderivatives

An antiderivative of a function $f(x)$ is a function $F(x)$ whose derivative is $f(x)$. That is, $F'(x) = f(x)$.

Example 1.1.1:

If $f(x) = 2x$, then $F(x) = x^2$ is an antiderivative because $\frac{d}{dx}(x^2) = 2x$.

However, $F(x) = x^2 + 5$ is also an antiderivative, as is $F(x) = x^2 - 100$. This leads to the concept of the constant of integration.

1.2 Indefinite Integrals

The set of all antiderivatives of $f(x)$ is called the indefinite integral of $f(x)$, denoted by $\int f(x) dx$.

$\int f(x) dx = F(x) + C$

Where:

• $\int$ is the integral sign.
• $f(x)$ is the integrand.
• $dx$ indicates that we are integrating with respect to $x$.
• $F(x)$ is any particular antiderivative of $f(x)$.
• $C$ is the constant of integration. This constant accounts for the fact that the derivative of any constant is zero.

Example 1.2.1:

$\int 2x dx = x^2 + C$

1.3 Definite Integrals (Area Under a Curve)

A definite integral computes the net signed area between the graph of a function $f(x)$ and the x-axis over a specified interval $[a, b]$.

$\int_a^b f(x) dx$

Where:

• $a$ is the lower limit of integration.
• $b$ is the upper limit of integration.

Unlike indefinite integrals, definite integrals evaluate to a numerical value, not a function.

1.4 The Fundamental Theorem of Calculus (FTC)

The FTC is a cornerstone of calculus, linking differentiation and integration. It has two parts, but the most commonly applied part for evaluating definite integrals is:

If $F(x)$ is an antiderivative of $f(x)$, then:

$\int_a^b f(x) dx = F(b) - F(a)$

Example 1.4.1:

Evaluate $\int_1^3 2x dx$

Chapter 2: Basic Integration Rules

Just like with derivatives, there are rules to simplify the process of finding antiderivatives.

2.1 Power Rule for Integration

If $n \neq -1$, then:

$\int x^n dx = \frac{x^{n+1}}{n+1} + C$

Example: $\int x^4 dx = \frac{x^{4+1}}{4+1} + C = \frac{x^5}{5} + C$

Example: $\int \sqrt{x} dx = \int x^{1/2} dx = \frac{x^{1/2+1}}{1/2+1} + C = \frac{x^{3/2}}{3/2} + C = \frac{2}{3}x^{3/2} + C$

2.2 Special Case: Integral of $\frac{1}{x}$

The power rule does not apply when $n=-1$. The integral of $\frac{1}{x}$ is the natural logarithm.

$\int \frac{1}{x} dx = \ln|x| + C$

Example: $\int \frac{5}{x} dx = 5 \int \frac{1}{x} dx = 5 \ln|x| + C$

2.3 Constant Multiple Rule

You can pull a constant factor out of the integral sign.

$\int cf(x) dx = c \int f(x) dx$

Example: $\int 3x^2 dx = 3 \int x^2 dx = 3 \left(\frac{x^3}{3}\right) + C = x^3 + C$

2.4 Sum and Difference Rules

The integral of a sum or difference of functions is the sum or difference of their integrals.

$\int (f(x) \pm g(x)) dx = \int f(x) dx \pm \int g(x) dx$

Example: $\int (4x^3 - 2x + 5) dx = \int 4x^3 dx - \int 2x dx + \int 5 dx$

$ = x^4 - x^2 + 5x + C$

2.5 Integrals of Common Functions

• Exponential Functions:

  $\int e^x dx = e^x + C$

  $\int a^x dx = \frac{a^x}{\ln a} + C$

• Trigonometric Functions:

  $\int \cos x dx = \sin x + C$

  $\int \sin x dx = -\cos x + C$

  $\int \sec^2 x dx = \tan x + C$

  $\int \csc^2 x dx = -\cot x + C$

  $\int \sec x \tan x dx = \sec x + C$

  $\int \csc x \cot x dx = -\csc x + C$

Chapter 3: Techniques of Integration

For more complex functions, we need specific techniques to find their integrals.

3.1 Substitution (U-Substitution)

This is the inverse of the Chain Rule. It helps integrate composite functions by simplifying the integrand.

Steps:

1. Choose a substitution $u = g(x)$, typically the "inside" function.
2. Calculate $du = g'(x) dx$.
3. Rewrite the integral entirely in terms of $u$ and $du$.
4. Integrate with respect to $u$.
5. Substitute back $g(x)$ for $u$ to express the answer in terms of $x$.

Example 3.1.1:

3.2 Integration by Parts (Brief Introduction)

This technique is the inverse of the Product Rule. It's used for integrals of products of functions that don't simplify well with substitution.

$\int u dv = uv - \int v du$

Choosing $u$ and $dv$ correctly is key to this method.

Example: $\int x e^x dx$ (This would be solved using integration by parts, which is a more advanced topic).

Chapter 4: Applications of Integrals

Integrals are crucial for solving problems involving accumulation, total change, and geometric measurements.

4.1 Area Under a Curve

The definite integral $\int_a^b f(x) dx$ represents the net signed area between the curve $y=f(x)$ and the x-axis from $x=a$ to $x=b$. Area above the x-axis is positive, below is negative.

Example 4.1.1:

Find the area under $f(x) = x^2$ from $x=0$ to $x=2$.

4.2 Displacement and Total Distance from Velocity

• If $v(t)$ is the velocity function of an object:
  • Displacement: $\int_{t_1}^{t_2} v(t) dt$ (Net change in position).
  • Total Distance Traveled: $\int_{t_1}^{t_2} |v(t)| dt$ (Sum of all path lengths, regardless of direction).

Example 4.2.1:

A particle's velocity is $v(t) = 2t - 4$ m/s. Find its displacement from $t=0$ to $t=3$ seconds.

4.3 Average Value of a Function

The average value of a function $f(x)$ over an interval $[a, b]$ is given by:

$\text{Average Value} = \frac{1}{b-a} \int_a^b f(x) dx$

Example 4.3.1:

Find the average value of $f(x) = x^2$ on the interval $[0, 3]$.

Chapter 5: Common Pitfalls and Tips for Success

Integration can be more challenging than differentiation. Be mindful of these common errors:

• Forgetting the "+ C": For indefinite integrals, always remember to add the constant of integration.
• Algebra Errors: Simplifying the integrand before integrating is crucial. Don't integrate term by term if there's a product or quotient unless it's a simple sum/difference.
• Misapplying Power Rule: Remember the special case for $\int x^{-1} dx$ ($\ln|x|$).
• U-Substitution Mistakes: Incorrectly choosing $u$, forgetting to find $du$, or not converting all terms (including limits for definite integrals) to $u$.
• Sign Errors: Especially with trigonometric integrals (e.g., $\int \sin x dx = -\cos x + C$).
• Absolute Value in Logarithms: Remember $\ln|x|$ for $\int \frac{1}{x} dx$.

Tips for Success:

• Check by Differentiating: The best way to check an indefinite integral is to differentiate your answer. If you get the original integrand, you're correct.
• Practice Basic Rules: Master the power rule and integrals of common functions before moving to techniques.
• Recognize Patterns: With practice, you'll start to recognize when u-substitution or other techniques are appropriate.
• Simplify the Integrand: Expand products, divide terms, or use trigonometric identities before integrating.
• Understand the Geometry: For definite integrals, visualize the area you are calculating.

Practice Problems (with Solutions)