Problem 47 - Calculus Early Transcendentals

Question

Determine the following indefinite integrals. Check your work by differentiation. $$\int \frac{1}{2 y} d y$$

Step-by-Step Solution

Answer

Question: Integrate the function $$\int \frac{1}{2 y} d y$$ and verify the result by differentiation. Answer: The indefinite integral of the given function is $$\frac{1}{2}\ln|y| + C_2$$, where C_2 is the constant of integration. Differentiating this result gives us $$\frac{1}{2}\frac{1}{y}$$, which matches the original function, confirming that our integration is correct.

Step 1: Integrate the given function

We have the function $$\int \frac{1}{2 y} d y$$ which can be simplified to $$\int \frac{1}{2} \cdot \frac{1}{y} d y$$. Now we can integrate the function with respect to y: $$\frac{1}{2} \int \frac{1}{y} d y$$

Step 2: Apply the Power Rule

Using the power rule, the integral of a function in the form of $$y^n$$ is $$\frac{y^{n+1}}{n+1} + C$$, where C is the constant of integration. In this case, we have $$y^{-1}$$, so n = -1. Applying the power rule, we have: $$\frac{1}{2} \cdot \frac{y^{-1+1}}{-1+1} + C = \frac{1}{2}(\ln|y| + C_1)$$

Step 3: Combine the constants

We can combine the constants $$\frac{1}{2}C_1$$ and C to form a new constant, C_2: $$\frac{1}{2}\ln|y| + C_2$$ So, our indefinite integral is: $$\int \frac{1}{2 y} d y = \frac{1}{2}\ln|y| + C_2$$

Step 4: Differentiate the result

To check the solution, let's differentiate the integral with respect to y: $$\frac{d}{dy}(\frac{1}{2}\ln|y| + C_2)$$ Using the chain rule, we know that the derivative of $$\ln|y|$$ with respect to y is $$\frac{1}{y}$$. So the differentiation becomes: $$\frac{1}{2}\frac{1}{y}$$ This matches the original function given in the exercise, so our integration is correct.