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Chapter 4: Problem 20

Find all the antiderivatives of the following functions. Check your work by taking derivatives. $$h(y)=y^{-1}$$

Short Answer

Expert verified
The antiderivatives of the function $$h(y) = y^{-1}$$ are given by $$\ln |y| + C$$, where $$C$$ is the constant of integration.

Step by step solution

01

Identify the integration formula for y^-1

We are given the function $$h(y) = y^{-1}$$. To find its antiderivative, we can refer to the table of standard integrals. We notice that $$\int y^m dy = \frac{y^{m+1}}{m+1} + C$$, where $$m$$ is a constant, and $$C$$ is the constant of integration. In our case, $$m = -1$$.
02

Apply the integration formula

Our function $$y^{-1}$$ is a special case, because the formula to find the antiderivative would require us to divide by $$m + 1 = -1 + 1 = 0$$ which is not possible. In this case, we have to use a natural logarithm as the antiderivative: $$\int y^{-1} dy = \ln |y| + C$$
03

Check the antiderivative by differentiating

Now we must differentiate the antiderivative that we found to see if it gives us the original function. Using the formula $$\frac{d}{dy}(\ln |y| + C)$$, we find that the derivative of the natural logarithm function is $$\frac{1}{y}$$, and the derivative of the constant $$C$$ is zero. Hence, the derivative is: $$\frac{d}{dy}(\ln |y| + C) = \frac{1}{y}$$ Since the derivative matches the original function $$y^{-1}$$, we have verified that the antiderivative $$\ln |y| + C$$ is correct.
04

Answer

The antiderivative of the function $$h(y) = y^{-1}$$ is $$\ln |y| + C$$.

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