Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
sin πx − 3sin 3x

Verified step by step guidance

1. Recall that an antiderivative (or indefinite integral) of a function \( f(x) \) is a function \( F(x) \) such that \( F'(x) = f(x) \). Our goal is to find \( F(x) \) given \( f(x) = \sin(\pi x) - 3 \sin(3x) \).
2. Use the basic antiderivative formula for sine: \( \int \sin(ax) \, dx = -\frac{1}{a} \cos(ax) + C \), where \( a \) is a constant and \( C \) is the constant of integration.
3. Apply this formula to each term separately: For \( \sin(\pi x) \), the antiderivative is \( -\frac{1}{\pi} \cos(\pi x) \). For \( -3 \sin(3x) \), factor out the constant \( -3 \) and integrate \( \sin(3x) \) to get \( -3 \times \left(-\frac{1}{3} \cos(3x)\right) \).
4. Simplify the expression after integration: The \( -3 \times -\frac{1}{3} \) simplifies to \( +1 \), so the antiderivative of \( -3 \sin(3x) \) is \( + \cos(3x) \).
5. Combine the results and add the constant of integration \( C \) to write the full antiderivative: \( F(x) = -\frac{1}{\pi} \cos(\pi x) + \cos(3x) + C \).

Key Concepts

Antiderivatives (Indefinite Integrals)

An antiderivative of a function is another function whose derivative equals the original function. Finding antiderivatives involves reversing differentiation, often represented as the indefinite integral with a constant of integration. For example, the antiderivative of sin(x) is -cos(x) + C.

Integration of Trigonometric Functions

Integrating trigonometric functions like sin(kx) requires using known integral formulas and applying the chain rule in reverse. Specifically, ∫sin(kx) dx = -cos(kx)/k + C, where k is a constant. Recognizing these patterns helps in quickly finding antiderivatives mentally.

Verification by Differentiation

After finding an antiderivative, differentiating it should return the original function. This step confirms the correctness of the solution. For example, differentiating -cos(πx)/π yields sin(πx), verifying the antiderivative.