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Chapter 4: Problem 47

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

Short Answer

Expert verified
Question: Find the antiderivative of the function \(\frac{1}{2y}\) with respect to \(y\), and verify your solution. Answer: The antiderivative of the function \(\frac{1}{2y}\) with respect to \(y\) is \(\frac{1}{2}\ln|y| + C\), where \(C\) is the constant of integration. Verification is obtained by differentiating the antiderivative to obtain the original function \(\frac{1}{2y}\).

Step by step solution

01

Find the antiderivative

To find the antiderivative of the function \(\frac{1}{2y}\) with respect to \(y\), we integrate it: $$\int \frac{1}{2y} dy$$ The integral of the given function can be written as: $$\frac{1}{2} \int \frac{1}{y} dy$$ Now, we know that the antiderivative of \(\frac{1}{y}\) with respect to \(y\) is \(\ln|y|\), so we can now write the result as: $$\frac{1}{2}\ln|y| + C$$ where \(C\) is the constant of integration.
02

Differentiate the antiderivative

To check our work, we will now differentiate the antiderivative function that we found: $$\frac{d}{dy}(\frac{1}{2}\ln|y| + C)$$ Applying the chain rule, we have: $$\frac{1}{2} \cdot \frac{1}{y} \cdot 1 + 0$$ Which simplifies to: $$\frac{1}{2y}$$ Since the derivative of our antiderivative function matches the original function, we have successfully verified our work. The correct antiderivative is: $$\frac{1}{2}\ln|y| + C$$

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Most popular questions from this chapter

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