What is the integral of (2x^2+4x-3)/(x^2) ?

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

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integral of (2x^2+4x-3)/(x^2)

\int \frac{2 x ^{2} + 4 x - 3}{x ^{2}} d x

2 x + 4 ln ∣ x ∣ + \frac{3}{x} + C

Solution steps

\int \frac{2 x ^{2} + 4 x - 3}{x ^{2}} d x

Expand

\frac{2 x ^{2} + 4 x - 3}{x ^{2}} : 2 + \frac{4}{x} - \frac{3}{x ^{2}}

= \int 2 + \frac{4}{x} - \frac{3}{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 2 d x + \int \frac{4}{x} d x - \int \frac{3}{x ^{2}} d x

\int 2 d x = 2 x

\int \frac{4}{x} d x = 4 ln ∣ x ∣

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

= 2 x + 4 ln ∣ x ∣ - (- \frac{3}{x})

Simplify

= 2 x + 4 ln ∣ x ∣ + \frac{3}{x}

Add a constant to the solution

= 2 x + 4 ln ∣ x ∣ + \frac{3}{x} + C

y^{^{''}} + 121 y = cos (10 t)

\int 2^{\frac{1}{2}}

\frac{\partial}{\partial x} (x^{3} y^{3})

x \to + 0 lim (\frac{x ^{3}}{ln ∣ x ∣})

sin (2 x) + cos (2 x)

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