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

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

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

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

\frac{x ^{3}}{3} - \frac{x ^{2}}{2} + 2 x - ln ∣ x ∣ + C

Solution steps

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

Expand

\frac{x ^{3} - ( x - 1 ) ^{2}}{x} : x^{2} - x + 2 - \frac{1}{x}

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

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

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

\int 2 d x = 2 x

\int \frac{1}{x} d x = ln ∣ x ∣

= \frac{x ^{3}}{3} - \frac{x ^{2}}{2} + 2 x - ln ∣ x ∣

Add a constant to the solution

= \frac{x ^{3}}{3} - \frac{x ^{2}}{2} + 2 x - ln ∣ x ∣ + C

x \to 1 lim (4 x - 1)

\frac{d}{d x} (100 x^{2} - 26 x e^{x})

\frac{d}{d x} (lo g_{10} (3 x + 4))

\frac{\partial}{\partial x} (tan (x y) + y e^{3 x})

y^{^{'}} + y = e^{- t} sin (t)

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