Problem 9 - AP Calculus with 8 Practice Tests 15th

Question

\(\int \cos 3 x d x=\) (A) \(3 \sin 3 x+C\) (B) \(-\sin 3 x+C\) (C) \(-\frac{1}{3} \sin 3 x+C\) (D) \(\frac{1}{3} \sin 3 x+C\)

Step-by-Step Solution

Answer

Option (D) \(\frac{1}{3} \sin 3 x+C\) is the correct answer.

Step 1: Identify the Integral Formula

We know that the integral of the cosine function is given by the formula: \[ \int \cos(kx) \, dx = \frac{1}{k} \sin(kx) + C \]This formula can be applied directly to the function we are given to integrate.

Step 2: Apply the Formula to the Given Function

In this exercise, the function inside the cosine is \(3x\), which means \(k = 3\). We use the formula from Step 1:\[ \int \cos(3x) \, dx = \frac{1}{3} \sin(3x) + C \]

Step 3: Confirm Your Solution

Compare the result \( \frac{1}{3} \sin(3x) + C \) with the options provided in the exercise.From the four options, (D) \(\frac{1}{3} \sin(3x) + C\) matches the solution derived from the integral formula.

Key Concepts

Trigonometric Integration

Trigonometric integration is a critical skill in calculus, especially when dealing with integrals involving trigonometric functions. Understanding how to integrate these functions helps solve a variety of physics and engineering problems.

• Common Trigonometric Integrals: Basic trigonometric integrals involve functions such as \(\sin(x)\), \(\cos(x)\), \(\tan(x)\), and others.
• Integration Formula: For example, the integral of \(\cos(kx)\) is \(\frac{1}{k} \sin(kx) + C\), where \(k\) is a constant.
• Integration by Parts: Occasionally, integration by parts or substitution might be necessary for more complex trigonometric expressions.

By mastering trigonometric integration, you are better prepared for tackling integrals in real-world contexts that involve oscillatory motion or wave phenomena. Regular practice will make these methods easier over time.

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivatives correspond to the integrand. You can think of them as the reverse process of differentiation. The result of an indefinite integral is not a single value but a function, possibly with an added constant.

• Not Defined by a Limit: Unlike indefinite integrals, definite integrals have specific limits.
• Basic Formula: If \(F(x)\) is an antiderivative of \(f(x)\), then \(\int f(x) \, dx = F(x) + C\).
• The Constant \(C\): The constant \(C\) represents all possible vertical shifts of the antiderivative function.

Understanding the concept of indefinite integrals is fundamental for solving problems involving rate of change, accumulated quantities, and much more. Always pay attention to the constant \(C\), as different initial conditions can yield different solutions.

Integral Calculus Steps

Understanding how to integrate a function involves several essential steps, especially for trigonometric functions or other complex expressions. Here’s a breakdown:

• Identify the Integral Type: Determine if you’re dealing with a basic trigonometric function or if substitution or integration by parts is necessary.
• Apply the Relevant Formula: Use the appropriate integration formula, such as familiar trigonometric rules for specific integrals.
• Consider the Constant: Always add the constant \(C\) to represent the indefinite integral.
• Verify the Solution: Compare your integrated solution with known results or check by differentiation to see if you can obtain the original function.

Mastering these steps will help you tackle a wide array of calculus problems confidently. Each step ensures you not only find a solution but understand the process and logic behind it.