Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
-(1/2)x⁻³ᐟ²

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Identify the function to find the antiderivative of: \(-\frac{1}{2} x^{-\frac{3}{2}}\).

Recall the power rule for antiderivatives: For \(x^n\), the antiderivative is \(\frac{x^{n+1}}{n+1} + C\), provided \(n \neq -1\).

Apply the power rule by increasing the exponent by 1: \(-\frac{3}{2} + 1 = -\frac{1}{2}\).

Write the antiderivative as \(-\frac{1}{2} \cdot \frac{x^{-\frac{1}{2}}}{-\frac{1}{2}} + C\).

Simplify the expression and add the constant of integration \(C\) to complete the antiderivative.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Antiderivatives (Indefinite Integrals)

An antiderivative of a function is another function whose derivative equals the original function. Finding antiderivatives involves reversing differentiation, often using known integral formulas. The result includes a constant of integration since differentiation of a constant is zero.

Power Rule for Integration

The power rule for integration states that ∫x^n dx = (x^(n+1))/(n+1) + C, for any real number n ≠ -1. This rule is essential for integrating functions with variable exponents, including negative and fractional powers, by increasing the exponent by one and dividing by the new exponent.

Verification by Differentiation

After finding an antiderivative, differentiating it should return the original function. This step confirms the correctness of the antiderivative and helps identify any mistakes in the integration process.

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