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

The integral of ((2+x)/x)^2 is -4/x+4ln|x|+x+C

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

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

- \frac{4}{x} + 4 ln ∣ x ∣ + x + C

Solution steps

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

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

Expand

\frac{( 2 + x ) ^{2}}{x ^{2}} : \frac{4}{x ^{2}} + \frac{4}{x} + 1

= \int \frac{4}{x ^{2}} + \frac{4}{x} + 1 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{4}{x ^{2}} d x + \int \frac{4}{x} d x + \int 1 d x

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

\int \frac{4}{x} d x = 4 ln ∣ x ∣

\int 1 d x = x

= - \frac{4}{x} + 4 ln ∣ x ∣ + x

Add a constant to the solution

= - \frac{4}{x} + 4 ln ∣ x ∣ + x + C

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What is the integral of ((2+x)/x)^2 ?
The integral of ((2+x)/x)^2 is -4/x+4ln|x|+x+C