What is the integral of (x/2-1/x+3/(x^3)-4/(x^4)) ?

The integral of (x/2-1/x+3/(x^3)-4/(x^4)) is (x^2)/4-ln|x|-3/(2x^2)+4/(3x^3)+C

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integral of (x/2-1/x+3/(x^3)-4/(x^4))

\int (\frac{x}{2} - \frac{1}{x} + \frac{3}{x ^{3}} - \frac{4}{x ^{4}}) d x

\frac{x ^{2}}{4} - ln ∣ x ∣ - \frac{3}{2 x ^{2}} + \frac{4}{3 x ^{3}} + C

Solution steps

\int \frac{x}{2} - \frac{1}{x} + \frac{3}{x ^{3}} - \frac{4}{x ^{4}} 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{x}{2} d x - \int \frac{1}{x} d x + \int \frac{3}{x ^{3}} d x - \int \frac{4}{x ^{4}} d x

\int \frac{x}{2} d x = \frac{x ^{2}}{4}

\int \frac{1}{x} d x = ln ∣ x ∣

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

\int \frac{4}{x ^{4}} d x = - \frac{4}{3 x ^{3}}

= \frac{x ^{2}}{4} - ln ∣ x ∣ - \frac{3}{2 x ^{2}} - (- \frac{4}{3 x ^{3}})

Simplify

= \frac{x ^{2}}{4} - ln ∣ x ∣ - \frac{3}{2 x ^{2}} + \frac{4}{3 x ^{3}}

Add a constant to the solution

= \frac{x ^{2}}{4} - ln ∣ x ∣ - \frac{3}{2 x ^{2}} + \frac{4}{3 x ^{3}} + C

What is the integral of (x/2-1/x+3/(x^3)-4/(x^4)) ?
The integral of (x/2-1/x+3/(x^3)-4/(x^4)) is (x^2)/4-ln|x|-3/(2x^2)+4/(3x^3)+C