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

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Identify the given function: \(-\pi \sin \pi x\).

Recall that the antiderivative (indefinite integral) of \(\sin(ax)\) is \(-\frac{1}{a} \cos(ax) + C\), where \(a\) is a constant and \(C\) is the constant of integration.

Apply this rule to the function \(-\pi \sin \pi x\): factor out the constant \(-\pi\) and integrate \(\sin \pi x\).

The integral becomes \(-\pi \int \sin \pi x \, dx = -\pi \left(-\frac{1}{\pi} \cos \pi x \right) + C\).

Simplify the expression by canceling \(\pi\) terms and write the antiderivative in its simplest form.

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주요 개념

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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 indefinite integrals with a constant of integration. For example, the antiderivative of cos(x) is sin(x) + C.

Differentiation of Trigonometric Functions

Understanding how to differentiate sine and cosine functions is essential to verify antiderivatives. For instance, the derivative of sin(πx) is πcos(πx), and the derivative of cos(πx) is -πsin(πx). This knowledge helps confirm if the found antiderivative is correct.

Constant Multiples in Integration

When integrating functions multiplied by constants, the constant can be factored out of the integral. For example, ∫ -πsin(πx) dx = -π ∫ sin(πx) dx. This simplifies the integration process and helps in finding the antiderivative efficiently.

Finding AntiderivativesIn Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.−πsin πx