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/(3³√x)

검증된 단계별 안내

Rewrite the given function in a form that is easier to integrate. Recall that the cube root of x is \(x^{1/3}\), so the function \(\frac{1}{3\sqrt[3]{x}}\) can be written as \(\frac{1}{3x^{1/3}}\).

Express the function as \(\frac{1}{3} x^{-1/3}\) by bringing the \(x^{1/3}\) in the denominator to the numerator with a negative exponent.

Use the power rule for antiderivatives, which states that 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 to \(\frac{1}{3} x^{-1/3}\) by increasing the exponent by 1: \(-\frac{1}{3} + 1 = \frac{2}{3}\), and then divide by the new exponent \(\frac{2}{3}\).

Multiply the constant \(\frac{1}{3}\) by the reciprocal of the new exponent \(\frac{3}{2}\) to find the coefficient of the antiderivative, and don't forget to 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 derivatives of constants are zero.

Power Rule for Integration

The power rule states that the integral of x^n with respect to x is (x^(n+1))/(n+1) + C, for any real number n ≠ -1. This rule is essential for integrating functions expressed as powers of x, including fractional and negative exponents.

Simplifying Expressions with Radicals and Exponents

Radicals can be rewritten as fractional exponents to simplify integration. For example, the cube root of x is x^(1/3). Converting radicals to exponents allows the use of the power rule directly and makes mental calculation easier.

Finding AntiderivativesIn Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.1/(3³√x)