23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.


∫ (1/2y)dy

Verified step by step guidance

1. Step 1: Recognize the integral ∫ (1/2y) dy as a basic indefinite integral. The goal is to find the antiderivative of the function (1/2y) with respect to y.
2. Step 2: Rewrite the integrand for clarity. The constant 1/2 can be factored out of the integral: ∫ (1/2y) dy = (1/2) ∫ (1/y) dy.
3. Step 3: Recall the antiderivative rule for 1/y. The integral of 1/y with respect to y is ln|y|, where ln represents the natural logarithm and the absolute value ensures the domain is valid.
4. Step 4: Combine the constant factor with the antiderivative. The result of the integral becomes (1/2) ln|y| + C, where C is the constant of integration.
5. Step 5: Verify your result by differentiating (1/2) ln|y| + C with respect to y. The derivative of (1/2) ln|y| is (1/2)(1/y), which matches the original integrand, confirming the solution is correct.

Key Concepts

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is often referred to as antiderivation, where we seek a function F(y) such that F'(y) equals the integrand.

Power Rule for Integration

The power rule for integration is a fundamental technique used to find indefinite integrals of polynomial functions. It states that the integral of y^n with respect to y is (y^(n+1))/(n+1) + C, where n is not equal to -1. This rule simplifies the process of integrating functions by allowing us to increase the exponent and divide by the new exponent.

Verification by Differentiation

Verification by differentiation involves checking the correctness of an indefinite integral by differentiating the resulting function. If the derivative of the antiderivative matches the original integrand, the integration is confirmed to be correct. This step is crucial in calculus to ensure that the integration process has been performed accurately.