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

\int 3 sin (t) d t

\int x \cdot (sin (x))^{2} d x

\int 3 sin^{2} (t) d t

y^{^{'''}} + 6 y^{^{''}} + 11 y^{^{'}} + 6 y = 0

\int \frac{e ^{2 x} + e ^{4 x}}{e ^{x} - e ^{5 x}} d x

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