What is the integral of (2x^{1/2}+3x^2)/(sqrt(x^3)) ?

The integral of (2x^{1/2}+3x^2)/(sqrt(x^3)) is 2ln|x|+2x^{3/2}+C

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integral of (2x^{1/2}+3x^2)/(sqrt(x^3))

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

2 ln ∣ x ∣ + 2 x^{\frac{3}{2}} + C

Solution steps

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

x^{3} = x^{\frac{3}{2}},

assuming

x \geq 0

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

Expand

\frac{2 x ^{\frac{1}{2}} + 3 x ^{2}}{x ^{\frac{3}{2}}} : \frac{2}{x} + 3 x^{\frac{1}{2}}

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

= 2 ln ∣ x ∣ + 2 x^{\frac{3}{2}}

Add a constant to the solution

= 2 ln ∣ x ∣ + 2 x^{\frac{3}{2}} + C

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What is the integral of (2x^{1/2}+3x^2)/(sqrt(x^3)) ?
The integral of (2x^{1/2}+3x^2)/(sqrt(x^3)) is 2ln|x|+2x^{3/2}+C