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

검증된 단계별 안내

Identify the integral to solve: $\int \left(8y - \frac{2}{y^{1/4}}\right) \, dy$.

Rewrite the integral by expressing the terms with exponents: $\int \left(8y - 2y^{-1/4}\right) \, dy$.

Split the integral into two separate integrals: $\int 8y \, dy - \int 2y^{-1/4} \, dy$.

Apply the power rule for integration to each term separately. Recall that $\int y^n \, dy = \frac{y^{n+1}}{n+1} + C$ for $n \neq -1$.

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.∫(8y − 2 / y¹ᐟ⁴) dy