Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
-(1/3)x⁻⁴ᐟ³

검증된 단계별 안내

Identify the function to find the antiderivative of: \(-\frac{1}{3} x^{-\frac{4}{3}}\).

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

Apply the power rule by adding 1 to the exponent: \(-\frac{4}{3} + 1 = -\frac{4}{3} + \frac{3}{3} = -\frac{1}{3}\).

Divide the original coefficient by the new exponent: Multiply \(-\frac{1}{3}\) by \(\frac{1}{-\frac{1}{3}}\) which is the reciprocal of the new exponent.

Write the antiderivative as \(F(x) = -\frac{1}{3} \times \frac{x^{-\frac{1}{3}}}{-\frac{1}{3}} + C\), then simplify the expression and add the constant of integration \(C\).

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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 using basic integration rules. The result includes a constant of integration since differentiation of a constant is zero.

Power Rule for Integration

The power rule states that the integral of x^n (where n ≠ -1) is (x^(n+1)) / (n+1) plus a constant. 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.

Finding AntiderivativesIn Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.-(1/3)x⁻⁴ᐟ³