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

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

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

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

\frac{x ^{2}}{2} + 2 x + ln ∣ x ∣ + C

Solution steps

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

Expand

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

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

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

\int 2 d x = 2 x

\int \frac{1}{x} d x = ln ∣ x ∣

= \frac{x ^{2}}{2} + 2 x + ln ∣ x ∣

Add a constant to the solution

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

sin (3 x)

y = e^{x} + 1

\frac{d}{d x} (- 2 sin (2 x) + 2 cos (2 x))

\frac{x}{x - 8}

x \to 0 + lim (2 x e^{- 3 x})

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