Checking Antiderivative Formulas

Verify the formulas in Exercises 57–62 by differentiation.

∫(3x + 5)⁻² dx = −(3x + 5)⁻¹/3 + C

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Identify the given antiderivative formula: \(\int (3x + 5)^{-2} \, dx = -\frac{(3x + 5)^{-1}}{3} + C\).

To verify this formula, differentiate the right-hand side expression \(-\frac{(3x + 5)^{-1}}{3} + C\) with respect to \(x\).

Apply the chain rule for differentiation: if \(f(x) = (3x + 5)^{-1}\), then \(f'(x) = -1 \cdot (3x + 5)^{-2} \cdot 3\) because the derivative of the inside function \(3x + 5\) is 3.

Multiply the derivative by the constant factor \(-\frac{1}{3}\) outside the function, simplifying the expression step-by-step.

Confirm that the resulting derivative simplifies exactly to the original integrand \((3x + 5)^{-2}\), which verifies the antiderivative formula.

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주요 개념

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

Antiderivatives and Indefinite Integrals

An antiderivative of a function is another function whose derivative equals the original function. Indefinite integrals represent the family of all antiderivatives and include a constant of integration, C, since differentiation of a constant is zero.

Power Rule for Integration

The power rule for integration states that ∫x^n dx = (x^(n+1))/(n+1) + C for any real number n ≠ -1. This rule helps integrate functions with variables raised to a power, including negative exponents, by increasing the exponent by one and dividing by the new exponent.

Verification by Differentiation

To verify an antiderivative formula, differentiate the proposed antiderivative and check if it equals the original integrand. This process confirms the correctness of the integral by applying the fundamental theorem of calculus in reverse.

Checking Antiderivative FormulasVerify the formulas in Exercises 57–62 by differentiation.∫(3x + 5)⁻² dx = −(3x + 5)⁻¹/3 + C