Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫x⁻¹ᐟ³ dx

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Recognize that the integral is of the form \(\int x^{m} \, dx\) where the exponent \(m\) is \(-\frac{1}{3}\).

Recall the power rule for integration: for any \(m \neq -1\), \(\int x^{m} \, dx = \frac{x^{m+1}}{m+1} + C\) where \(C\) is the constant of integration.

Calculate the new exponent by adding 1 to \(m\): \(m + 1 = -\frac{1}{3} + 1 = \frac{2}{3}\).

Apply the power rule formula: \(\int x^{-\frac{1}{3}} \, dx = \frac{x^{\frac{2}{3}}}{\frac{2}{3}} + C\).

Simplify the fraction in the denominator by multiplying numerator and denominator appropriately, and remember to add the constant of integration \(C\).

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Indefinite Integral and Antiderivative

An indefinite integral represents the most general form of the antiderivative of a function, including a constant of integration. It reverses differentiation, finding a function whose derivative matches the integrand. For example, ∫f(x) dx = F(x) + C, where F'(x) = f(x).

Power Rule for Integration

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

Handling Fractional and Negative Exponents

When integrating expressions like x raised to a fractional or negative exponent, treat the exponent as a rational number. Apply the power rule carefully, ensuring the exponent is not -1, and simplify the result. For example, x^(-1/3) integrates to (x^(2/3))/(2/3) + C.

Finding Indefinite IntegralsIn Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.∫x⁻¹ᐟ³ dx