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Chapter 4: Problem 33

Determine the following indefinite integrals. Check your work by differentiation. $$\int\left(\frac{3}{x^{4}}+2-\frac{3}{x^{2}}\right) d x$$

Short Answer

Expert verified
Question: Find the indefinite integral of the function $$\int\left(\frac{3}{x^{4}}+2-\frac{3}{x^{2}}\right) d x$$ Answer: The indefinite integral of the given function is $$-\frac{3}{x^3} + 2x + \frac{3}{x} + C$$

Step by step solution

01

Rewrite the function in a more convenient form

First, let's rewrite the given function to make it easier to work with. The given function is: $$\int\left(\frac{3}{x^{4}}+2-\frac{3}{x^{2}}\right) d x$$ We can rewrite it as: $$\int\left(3x^{-4}+2+(-3x^{-2})\right) d x$$
02

Calculating the indefinite integral of each term

Now, we'll calculate the indefinite integral for each term separately using the power rule for integration: 1. Calculate the indefinite integral of \(3x^{-4}\): $$\int(3x^{-4})dx$$ Using the power rule: $$\frac{3x^{-3}}{-3}+C_1$$ 2. Calculate the indefinite integral of 2: $$\int(2)dx$$ Using the power rule: $$2x+C_2$$ 3. Calculate the indefinite integral of \(-3x^{-2}\): $$\int(-3x^{-2})dx$$ Using the power rule: $$\frac{-3x^{-1}}{-1}+C_3$$
03

Combine the results

Now, let's combine the results from the previous step: $$\int\left(3x^{-4}+2+(-3x^{-2})\right) d x = \frac{3x^{-3}}{-3}+C_1 + 2x+C_2 +\frac{-3x^{-1}}{-1}+C_3$$ Combine the constants C_1, C_2, and C_3 into a single constant C: $$-\frac{3}{x^3} + 2x + \frac{3}{x} + C$$ Thus, the indefinite integral of the given function is: $$\int\left(\frac{3}{x^{4}}+2-\frac{3}{x^{2}}\right) d x=-\frac{3}{x^3} + 2x + \frac{3}{x} + C$$
04

Check the work by differentiation

Now, we'll differentiate our result to check our work: $$\frac{d}{dx}\left(-\frac{3}{x^3} + 2x + \frac{3}{x} + C\right)$$ Differentiate each term separately: 1. Differentiate \(-\frac{3}{x^3}\): $$-\frac{-9}{x^4}$$ 2. Differentiate \(2x\): $$2$$ 3. Differentiate \(\frac{3}{x}\): $$-3x^{-2}$$ 4. Differentiate the constant C: $$0$$ Combine the results: $$-\frac{-9}{x^4} + 2 + (-3x^{-2})$$ Simplify the expression: $$\frac{9}{x^4} + 2 - \frac{3}{x^2}$$ This is the same as the original function in the given exercise. Hence, our indefinite integral is correct.

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