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Integrate the function $\frac{8}{x}-\frac{5}{x^{2}}+\frac{6}{x^{3}}$ with respect to $x$ .

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Evaluate integers raised to rational exponents

Full solution

Q. Integrate the function

\frac{8}{x}-\frac{5}{x^{2}}+\frac{6}{x^{3}}

with respect to

x

.

Given function: We are given the function to integrate: $(\frac{8}{x}) - (\frac{5}{x^2}) + (\frac{6}{x^3})$ . We will integrate each term separately.
Integrate first term: Integrate the first term $\frac{8}{x}$ with respect to $x$ . The integral of $\frac{1}{x}$ with respect to $x$ is $\ln|x|$ . Therefore, the integral of $\frac{8}{x}$ is $8\ln|x|$ .
Integrate second term: Integrate the second term $-\frac{5}{x^2}$ with respect to $x$ . The integral of $\frac{1}{x^n}$ with respect to $x$ is $-\frac{1}{(n-1)x^{(n-1)}}$ for $n \neq 1$ . Here, $n=2$ , so the integral of $-\frac{5}{x^2}$ is $\frac{5}{x}$ .
Integrate third term: Integrate the third term $\frac{6}{x^3}$ with respect to $x$ . Using the same rule as in the previous step, the integral of $\frac{6}{x^3}$ is $-\frac{3}{x^2}$ .
Combine results: Combine the results of the three integrals. $\newline$ The integral of the entire function is $8\ln|x| + \frac{5}{x} - \frac{3}{x^2} + C$ , where $C$ is the constant of integration.

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