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

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

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

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

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

Solution steps

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

Expand

\frac{6 x ^{5} - x + 5 x ^{2}}{x ^{3}} : 6 x^{2} - \frac{1}{x ^{\frac{5}{2}}} + \frac{5}{x}

= \int 6 x^{2} - \frac{1}{x ^{\frac{5}{2}}} + \frac{5}{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 6 x^{2} d x - \int \frac{1}{x ^{\frac{5}{2}}} d x + \int \frac{5}{x} d x

\int 6 x^{2} d x = 2 x^{3}

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

\int \frac{5}{x} d x = 5 ln ∣ x ∣

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

Simplify

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

Add a constant to the solution

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

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