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

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

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

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

x + 2 ln ∣ x ∣ - \frac{1}{x} + C

Solution steps

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

Expand

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

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

\int 1 d x = x

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

= x + 2 ln ∣ x ∣ - \frac{1}{x}

Add a constant to the solution

= x + 2 ln ∣ x ∣ - \frac{1}{x} + C

sin (x), 3

x \to 2 lim (4 x + 1)

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

\int e^{x^{3}} d x

x \to - 3 lim (x^{2} + 7 x + 12)

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