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

The integral of ((y^2+1)^2)/(y^2) is (y^3)/3+2y-1/y+C

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

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

\frac{y ^{3}}{3} + 2 y - \frac{1}{y} + C

Solution steps

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

Expand

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

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

\int 2 d y = 2 y

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

= \frac{y ^{3}}{3} + 2 y - \frac{1}{y}

Add a constant to the solution

= \frac{y ^{3}}{3} + 2 y - \frac{1}{y} + C

x \to 2 lim (\frac{3}{x})

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

\int 3 x^{2} y + e^{x} sin (y) d x

y^{^{'}} + \frac{( x - 2 )}{x - 1} y = 0

\frac{d}{d x} (x^{2} \cdot ln (2 x))

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