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Chapter 5: Problem 88

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{1}^{2}\left(\frac{2}{s}-\frac{4}{s^{3}}\right) d s$$

Short Answer

Expert verified
Question: Evaluate the definite integral: $$\int_{1}^{2}\left(\frac{2}{s}-\frac{4}{s^{3}}\right) d s$$ Answer: $$\int_{1}^{2}\left(\frac{2}{s}-\frac{4}{s^{3}}\right) d s = 2 \ln 2 -\frac{1}{2} + 2$$

Step by step solution

01

Identify the given function inside the integral

First, we need to identify the function which is inside the integral: $$f(s)=\frac{2}{s}-\frac{4}{s^{3}}$$
02

Find the antiderivative F(s) of the given function f(s)

We can find the antiderivative of the function f(s) by doing the following: $$F(s)=\int (\frac{2}{s}-\frac{4}{s^{3}})d s=\int \frac{2}{s} ds - \int \frac{4}{s^{3}} ds$$ Now, we find the antiderivatives of each term separately: $$\int \frac{2}{s} ds = 2 \int \frac{1}{s} ds = 2 \ln |s| + C_1$$ $$\int \frac{4}{s^{3}} ds = 4 \int s^{-3} ds = -\frac{2}{s^2} + C_2$$ So, we combine the antiderivatives and constants: $$F(s) = 2 \ln |s| - \frac{2}{s^2} + C$$
03

Apply the Fundamental Theorem of Calculus to the antiderivative F(s) with the given limits

Applying the Fundamental Theorem of Calculus to the antiderivative F(s), we have: $$\int_{1}^{2}\left(\frac{2}{s}-\frac{4}{s^{3}}\right) d s= F(2)-F(1)$$ Substitute the limits into the antiderivative: $$F(2) = 2 \ln |2| - \frac{2}{2^2} = 2 \ln 2 - \frac{1}{2}$$ $$F(1) = 2 \ln |1| - \frac{2}{1^2} = 0 - 2$$
04

Calculate the definite integral

Now, subtract F(1) from F(2) to obtain the definite integral: $$\int_{1}^{2}\left(\frac{2}{s}-\frac{4}{s^{3}}\right) d s= F(2)-F(1) = (2 \ln 2 - \frac{1}{2}) - (-2) = 2 \ln 2 -\frac{1}{2} + 2$$ So, the definite integral is: $$\int_{1}^{2}\left(\frac{2}{s}-\frac{4}{s^{3}}\right) d s = 2 \ln 2 -\frac{1}{2} + 2$$

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