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

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

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integral of ((u^2+1)^2)/(u^3)

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

\frac{u ^{2}}{2} + 2 ln ∣ u ∣ - \frac{1}{2 u ^{2}} + C

Solution steps

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

Expand

\frac{( u ^{2} + 1 ) ^{2}}{u ^{3}} : u + \frac{2}{u} + \frac{1}{u ^{3}}

= \int u + \frac{2}{u} + \frac{1}{u ^{3}} 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 d u + \int \frac{2}{u} d u + \int \frac{1}{u ^{3}} d u

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

\int \frac{2}{u} d u = 2 ln ∣ u ∣

\int \frac{1}{u ^{3}} d u = - \frac{1}{2 u ^{2}}

= \frac{u ^{2}}{2} + 2 ln ∣ u ∣ - \frac{1}{2 u ^{2}}

Add a constant to the solution

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

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