Problem 11 - Thomas Calculus

Question

In Exercises \(1-16,\) find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation. $$\text { a. }-\pi \sin \pi x \quad \text { b. } 3 \sin x \quad \text { c. } \sin \pi x-3 \sin 3 x$$

Step-by-Step Solution

Answer

Antiderivatives: (a) \(\cos(\pi x) + C\), (b) \(-3 \cos x + C\), (c) \(-\frac{1}{\pi} \cos \pi x + \cos 3x + C\).

Step 1: Identify the Function to Differentiate

We need to find an antiderivative for each function given. Let's start with function (a), which is \[-\pi \sin \pi x\].

Step 2: Recognize the Basic Antiderivative Rule

Recall that the antiderivative of \(\sin(ax)\) is \[-\frac{1}{a}\cos(ax)+C\],where \(a\) is a constant and \(C\) is the constant of integration. We apply this rule to the function.

Step 3: Find Antiderivative for Function (a)

Using the rule from Step 2, \(a = \pi\), so the antiderivative for \(-\pi \sin \pi x\) is \[-\pi \left(-\frac{1}{\pi}\right) \cos(\pi x) = \cos(\pi x) + C\].

Step 4: Check Function (a) by Differentiation

Differentiate the antiderivative \(\cos(\pi x) + C\) to confirm it results in\(-\pi \sin \pi x\). The derivative of \(\cos(\pi x)\) is \(-\pi \sin \pi x\), which matches the original function.

Step 5: Find Antiderivative for Function (b)

Given: \(3 \sin x\). Using the antiderivative rule, the result is:\(-3 \cos x + C\),where \(C\) is the constant of integration.

Step 6: Verify Function (b) by Differentiation

Find the derivative of \(-3 \cos x + C\). The derivative is \(3 \sin x\), confirming our antiderivative is correct.

Step 7: Find Antiderivative for Function (c)

Given function:\(\sin \pi x - 3 \sin 3x\). Apply the antiderivative rule separately:- The antiderivative of \(\sin \pi x\)is \(-\frac{1}{\pi} \cos \pi x\).- The antiderivative of \(-3 \sin 3x\)is \((3 \times \frac{1}{3})(-\cos 3x) = \cos 3x\). Combine to get\(-\frac{1}{\pi} \cos \pi x + \cos 3x + C\).

Step 8: Confirm Function (c) by Differentiation

Differentiate the expression:\(-\frac{1}{\pi} \cos \pi x + \cos 3x + C\). This results in:\(\sin \pi x - 3 \sin 3 x\),which matches the original function.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that focuses on how functions change. In simpler terms, it helps us understand the rate at which one quantity is changing with respect to another. For example, if you have a function representing distance over time, differentiating this function gives you the velocity, or how fast the distance is changing over time.

Here's what you need to know about differentiation:

• Basic Principle: The process of differentiation involves finding the derivative of a function, which represents the rate of change.
• Notations: Common notations include \( f'(x)\) or \( \frac{dy}{dx}\), where \(y\) is a function of \(x\).
• Rules of Differentiation: Differentiation has specific rules like the power rule, product rule, and chain rule, which help simplify finding derivatives.

Differentiation is useful not only in mathematics but in various fields like physics, engineering, and economics, where understanding changes is crucial.

Integration Rules

Integration is often considered the reverse process of differentiation. Instead of finding how a function changes, we're interested in finding the original function given its rate of change. This process is known as finding an "antiderivative."

Understanding integration rules is crucial for solving calculus problems:

• Basic Antiderivative Rule: If the derivative of \(F(x)\) is \(f(x)\), then \(F(x)\) is an antiderivative of \(f(x)\). This is often expressed as finding \(\int f(x)\,dx = F(x) + C\), where \(C\) is the constant of integration.
• Constant Multiplier Rule: \(\int cf(x)\,dx = c\int f(x)\,dx\). This means you can factor out constants from the integral.
• Sum Rule: \(\int [f(x) + g(x)]\,dx = \int f(x)\,dx + \int g(x)\,dx\). You can break up the integral of a sum into separate integrals.

The steps taken in the original exercise utilize these integration rules to find antiderivatives effectively, helping you break down complex expressions into simpler parts.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate angles of a triangle to the lengths of its sides. The three primary trigonometric functions are sine, cosine, and tangent, often abbreviated as sin, cos, and tan. These functions are cyclic and model periodic phenomena, making them vital in fields such as physics and engineering.

When dealing with trigonometric functions, there are a few integration rules you should be aware of:

• Sine Function: The antiderivative of \(\sin(ax)\) is \(-\frac{1}{a}\cos(ax) + C\). In our examples, this rule helps us find antiderivatives of functions involving sine.
• Cosine Function: On the flip side, the antiderivative of \(\cos(ax)\) is \(\frac{1}{a}\sin(ax) + C\).
• Application: These rules help in the integration process, as seen in the exercises where functions like \(3\sin x\) or \(-\pi \sin \pi x\) require specific handling.

Understanding these functions and their properties aids in effectively applying integration techniques, especially when dealing with combinations of different trigonometric terms.