What is the integral of (3x^5-6x^2+2x-5)/(x^3) ?

The integral of (3x^5-6x^2+2x-5)/(x^3) is x^3-6ln|x|-2/x+5/(2x^2)+C

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integral of (3x^5-6x^2+2x-5)/(x^3)

\int \frac{3 x ^{5} - 6 x ^{2} + 2 x - 5}{x ^{3}} d x

x^{3} - 6 ln ∣ x ∣ - \frac{2}{x} + \frac{5}{2 x ^{2}} + C

Solution steps

\int \frac{3 x ^{5} - 6 x ^{2} + 2 x - 5}{x ^{3}} d x

Expand

\frac{3 x ^{5} - 6 x ^{2} + 2 x - 5}{x ^{3}} : 3 x^{2} - \frac{6}{x} + \frac{2}{x ^{2}} - \frac{5}{x ^{3}}

= \int 3 x^{2} - \frac{6}{x} + \frac{2}{x ^{2}} - \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

= \int 3 x^{2} d x - \int \frac{6}{x} d x + \int \frac{2}{x ^{2}} d x - \int \frac{5}{x ^{3}} d x

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

\int \frac{6}{x} d x = 6 ln ∣ x ∣

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

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

= x^{3} - 6 ln ∣ x ∣ - \frac{2}{x} - (- \frac{5}{2 x ^{2}})

Simplify

= x^{3} - 6 ln ∣ x ∣ - \frac{2}{x} + \frac{5}{2 x ^{2}}

Add a constant to the solution

= x^{3} - 6 ln ∣ x ∣ - \frac{2}{x} + \frac{5}{2 x ^{2}} + C

n = 0 \sum \infty e^{4 n}

\int_{2}^{- 1} 3^{x} d x

n = 2 \sum \infty \frac{1}{n ^{\frac{3}{2}}}

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

\frac{d}{d x} (\frac{1}{cos ( x )})

What is the integral of (3x^5-6x^2+2x-5)/(x^3) ?
The integral of (3x^5-6x^2+2x-5)/(x^3) is x^3-6ln|x|-2/x+5/(2x^2)+C