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

검증된 단계별 안내

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 \( f(x) = \sin(\pi x) - 3 \sin(3x) \).

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.

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 \( -3 \times \left(-\frac{1}{3} \cos(3x)\right) \).

Simplify the expression after integration: The \( -3 \times -\frac{1}{3} \) simplifies to \( +1 \), so the antiderivative of \( -3 \sin(3x) \) is \( + \cos(3x) \).

Combine the results and add the constant of integration \( C \) to write the full antiderivative: \( F(x) = -\frac{1}{\pi} \cos(\pi x) + \cos(3x) + C \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

Finding AntiderivativesIn 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