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

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

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

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

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

Solution steps

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

Expand

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

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

Apply the Sum Rule:

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

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

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

\int 2 d u = 2 u

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

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

Add a constant to the solution

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

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