What is the integral of ((2x+1)^3)/(x^2) ?

The integral of ((2x+1)^3)/(x^2) is 4x^2+12x+6ln|x|-1/x+C

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integral of ((2x+1)^3)/(x^2)

\int \frac{( 2 x + 1 ) ^{3}}{x ^{2}} d x

4 x^{2} + 12 x + 6 ln ∣ x ∣ - \frac{1}{x} + C

Solution steps

\int \frac{( 2 x + 1 ) ^{3}}{x ^{2}} d x

Expand

\frac{( 2 x + 1 ) ^{3}}{x ^{2}} : 8 x + 12 + \frac{6}{x} + \frac{1}{x ^{2}}

= \int 8 x + 12 + \frac{6}{x} + \frac{1}{x ^{2}} 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 8 x d x + \int 12 d x + \int \frac{6}{x} d x + \int \frac{1}{x ^{2}} d x

\int 8 x d x = 4 x^{2}

\int 12 d x = 12 x

\int \frac{6}{x} d x = 6 ln ∣ x ∣

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

= 4 x^{2} + 12 x + 6 ln ∣ x ∣ - \frac{1}{x}

Add a constant to the solution

= 4 x^{2} + 12 x + 6 ln ∣ x ∣ - \frac{1}{x} + C

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What is the integral of ((2x+1)^3)/(x^2) ?
The integral of ((2x+1)^3)/(x^2) is 4x^2+12x+6ln|x|-1/x+C