What is the integral of (x^2-5)/(2x^3) ?

The integral of (x^2-5)/(2x^3) is 1/2 (ln|x|+5/(2x^2))+C

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integral of (x^2-5)/(2x^3)

\int \frac{x ^{2} - 5}{2 x ^{3}} d x

\frac{1}{2} (ln ∣ x ∣ + \frac{5}{2 x ^{2}}) + C

Solution steps

\int \frac{x ^{2} - 5}{2 x ^{3}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{x ^{2} - 5}{x ^{3}} d x

Expand

\frac{x ^{2} - 5}{x ^{3}} : \frac{1}{x} - \frac{5}{x ^{3}}

= \frac{1}{2} \cdot \int \frac{1}{x} - \frac{5}{x ^{3}} 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

= \frac{1}{2} (\int \frac{1}{x} d x - \int \frac{5}{x ^{3}} d x)

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{5}{x ^{3}} d x = - \frac{5}{2 x ^{2}}

= \frac{1}{2} (ln ∣ x ∣ - (- \frac{5}{2 x ^{2}}))

Simplify

= \frac{1}{2} (ln ∣ x ∣ + \frac{5}{2 x ^{2}})

Add a constant to the solution

= \frac{1}{2} (ln ∣ x ∣ + \frac{5}{2 x ^{2}}) + C

What is the integral of (x^2-5)/(2x^3) ?
The integral of (x^2-5)/(2x^3) is 1/2 (ln|x|+5/(2x^2))+C