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.


∫(8y − 2 / y¹ᐟ⁴) dy

Verified step by step guidance

1. Identify the integral to solve: \(\int \left(8y - \frac{2}{y^{1/4}}\right) \, dy\).
2. Rewrite the integral by expressing the terms with exponents: \(\int \left(8y - 2y^{-1/4}\right) \, dy\).
3. Split the integral into two separate integrals: \(\int 8y \, dy - \int 2y^{-1/4} \, dy\).
4. Apply the power rule for integration to each term separately. Recall that \(\int y^n \, dy = \frac{y^{n+1}}{n+1} + C\) for \(n \neq -1\).
5. Integrate each term: For \$8y\(, treat it as \)8y^1\(, so the integral is \(8 \times \frac{y^{1+1}}{1+1}\). For \(2y^{-1/4}\), the integral is \(2 \times \frac{y^{-1/4 + 1}}{-1/4 + 1}\). Don't forget to add the constant of integration \)+ C$ at the end.

Key Concepts

Indefinite Integral and Antiderivative

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

Power Rule for Integration

The power rule states that ∫y^n dy = (y^(n+1)) / (n+1) + C for any real number n ≠ -1. This rule is essential for integrating terms like y raised to a power, including fractional and negative exponents, 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 integrand. This step confirms the correctness of the integral solution. Checking by differentiation helps identify and correct errors in the integration process.