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

The integral of ((4+x^2)^2)/(x^3) is -8/(x^2)+8ln|x|+(x^2)/2+C

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

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

- \frac{8}{x ^{2}} + 8 ln ∣ x ∣ + \frac{x ^{2}}{2} + C

Solution steps

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

Expand

\frac{( 4 + x ^{2} ) ^{2}}{x ^{3}} : \frac{16}{x ^{3}} + \frac{8}{x} + x

= \int \frac{16}{x ^{3}} + \frac{8}{x} + x 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 \frac{16}{x ^{3}} d x + \int \frac{8}{x} d x + \int x d x

\int \frac{16}{x ^{3}} d x = - \frac{8}{x ^{2}}

\int \frac{8}{x} d x = 8 ln ∣ x ∣

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

= - \frac{8}{x ^{2}} + 8 ln ∣ x ∣ + \frac{x ^{2}}{2}

Add a constant to the solution

= - \frac{8}{x ^{2}} + 8 ln ∣ x ∣ + \frac{x ^{2}}{2} + C

6 e^{- 3} - t^{2} + 2 t - 8

x y^{^{'}} + 2 y = - 2 sin (x)

\int \frac{48}{( 6 x + 1 ) ^{9}} d x

\int 10 sin^{3} (x co) s^{7} x d x

\frac{4 x}{x}

What is the integral of ((4+x^2)^2)/(x^3) ?
The integral of ((4+x^2)^2)/(x^3) is -8/(x^2)+8ln|x|+(x^2)/2+C