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 + ³√x) dx

Verified step by step guidance

1. Rewrite the integral in terms of exponents to make it easier to integrate. Recall that \( \sqrt{x} = x^{1/2} \) and \( \sqrt[3]{x} = x^{1/3} \). So the integral becomes \( \int \left( x^{1/2} + x^{1/3} \right) \, dx \).
2. Use the power rule for integration, which states that for any real number \( n \neq -1 \), \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.
3. Apply the power rule separately to each term inside the integral: \( \int x^{1/2} \, dx = \frac{x^{1/2 + 1}}{1/2 + 1} + C_1 = \frac{x^{3/2}}{3/2} + C_1 \) and \( \int x^{1/3} \, dx = \frac{x^{1/3 + 1}}{1/3 + 1} + C_2 = \frac{x^{4/3}}{4/3} + C_2 \).
4. Simplify the fractions in the denominators by multiplying numerator and denominator appropriately: \( \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2} \) and \( \frac{x^{4/3}}{4/3} = \frac{3}{4} x^{4/3} \).
5. Combine the results and write the most general antiderivative as \( \frac{2}{3} x^{3/2} + \frac{3}{4} x^{4/3} + C \), where \( C \) is the constant of integration. To verify, differentiate this expression and check if you get back the original integrand.

Key Concepts

Indefinite Integral (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 states that ∫x^n dx = (x^(n+1))/(n+1) + C for any real number n ≠ -1. This rule is essential for integrating terms like √x (x^(1/2)) and ³√x (x^(1/3)) by rewriting roots as fractional exponents and applying the formula.

Verification by Differentiation

After finding an antiderivative, differentiating it should return the original integrand. This step confirms the correctness of the integral solution. It ensures no algebraic errors occurred during integration and that the constant of integration is properly accounted for.