What is the integral of ((x+3)^2)/(x^5) ?

The integral of ((x+3)^2)/(x^5) is -1/(2x^2)-2/(x^3)-9/(4x^4)+C

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integral of ((x+3)^2)/(x^5)

\int \frac{( x + 3 ) ^{2}}{x ^{5}} d x

- \frac{1}{2 x ^{2}} - \frac{2}{x ^{3}} - \frac{9}{4 x ^{4}} + C

Solution steps

\int \frac{( x + 3 ) ^{2}}{x ^{5}} d x

Expand

\frac{( x + 3 ) ^{2}}{x ^{5}} : \frac{1}{x ^{3}} + \frac{6}{x ^{4}} + \frac{9}{x ^{5}}

= \int \frac{1}{x ^{3}} + \frac{6}{x ^{4}} + \frac{9}{x ^{5}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{x ^{3}} d x + \int \frac{6}{x ^{4}} d x + \int \frac{9}{x ^{5}} d x

\int \frac{1}{x ^{3}} d x = - \frac{1}{2 x ^{2}}

\int \frac{6}{x ^{4}} d x = - \frac{2}{x ^{3}}

\int \frac{9}{x ^{5}} d x = - \frac{9}{4 x ^{4}}

= - \frac{1}{2 x ^{2}} - \frac{2}{x ^{3}} - \frac{9}{4 x ^{4}}

Add a constant to the solution

= - \frac{1}{2 x ^{2}} - \frac{2}{x ^{3}} - \frac{9}{4 x ^{4}} + C

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What is the integral of ((x+3)^2)/(x^5) ?
The integral of ((x+3)^2)/(x^5) is -1/(2x^2)-2/(x^3)-9/(4x^4)+C