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Chapter 7: Problem 10

Use a table of integrals to determine the following indefinite integrals. $$\int \frac{d y}{y(2 y+9)}$$

Short Answer

Expert verified
The indefinite integral of the given function is $\int\frac{dy}{y(2y+9)} = \frac{1}{9}\ln|y| + \frac{1}{9}\ln|2y+9| + C$, where C is the constant of integration.

Step by step solution

01

Perform Partial Fraction Decomposition

First, we need to rewrite the given integrand as the sum of two simpler rational functions using partial fraction decomposition. To do so, we express the given integrand in the form: $$\frac{1}{y(2 y+9)} = \frac{A}{y} + \frac{B}{2 y+9}$$ Now, we need to solve for A and B.
02

Solve for A and B

To solve for A and B, we can first use the common denominator to make the equation as follows: $$1 = A(2y + 9) + By$$ Now, we can set y = 0 to solve for A: $$1 = A(9)$$ $$A = \frac{1}{9}$$ Now, let's set \(2y+9=0 \Rightarrow y=-\frac{9}{2}\): $$1 = \frac{9B}{2}$$ $$B = \frac{2}{9}$$ Now we have the values for A and B: $$\frac{1}{y(2 y+9)} = \frac{1}{9}\frac{1}{y} + \frac{2}{9}\frac{1}{2 y+9}$$
03

Integrate Each Term Separately

Now, we can rewrite the integral using the partial fraction decomposition: $$\int \frac{dy}{y(2 y+9)} = \int \frac{1}{9}\frac{1}{y}\, dy + \int \frac{2}{9}\frac{1}{2 y+9}\, dy$$ Since both of these integrals are in a more standard form, we can now evaluate them using a table of integrals. We know that \(\int \frac{1}{y}dy = \ln|y|\) and \(\int \frac{1}{ax+b} = \frac{1}{a}\ln|ax+b|\). So, applying these rules: $$\int \frac{1}{9}\frac{1}{y}\, dy = \frac{1}{9} \ln|y| + C_1$$ $$\int \frac{2}{9}\frac{1}{2 y+9}\, dy = \frac{2}{9}\cdot \frac{1}{2}\ln|2y+9| + C_2$$
04

Combine the Results and Solve for the Constant

Now, we can combine the integrals and simplify the result: $$\int \frac{dy}{y(2 y+9)} = \frac{1}{9}\ln|y| + \frac{1}{9}\ln|2y+9| + C$$ Where \(C = C_1 + C_2\) is the constant of integration. Therefore, the indefinite integral of \(\int\frac{dy}{y(2y+9)}\) is: $$\int\frac{dy}{y(2y+9)} = \frac{1}{9}\ln|y| + \frac{1}{9}\ln|2y+9| + C$$

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