What is the integral of ((x^3+3x+ln(2)))/(sqrt(x^3)) ?

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

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integral of ((x^3+3x+ln(2)))/(sqrt(x^3))

\int \frac{( x ^{3} + 3 x + ln ( 2 ) )}{x ^{3}} d x

\frac{2}{5} x^{\frac{5}{2}} + 6 x - \frac{2 ln ( 2 )}{x} + C

Solution steps

\int \frac{x ^{3} + 3 x + ln ( 2 )}{x ^{3}} d x

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

assuming

x \geq 0

= \int \frac{x ^{3} + 3 x + ln ( 2 )}{x ^{\frac{3}{2}}} d x

Expand

\frac{x ^{3} + 3 x + ln ( 2 )}{x ^{\frac{3}{2}}} : x^{\frac{3}{2}} + \frac{3}{x ^{\frac{1}{2}}} + \frac{ln ( 2 )}{x ^{\frac{3}{2}}}

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

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

\int \frac{3}{x ^{\frac{1}{2}}} d x = 6 x^{\frac{1}{2}}

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

= \frac{2}{5} x^{\frac{5}{2}} + 6 x^{\frac{1}{2}} - \frac{2 ln ( 2 )}{x}

Apply radical rule:

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

Add a constant to the solution

= \frac{2}{5} x^{\frac{5}{2}} + 6 x - \frac{2 ln ( 2 )}{x} + C

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

x (\frac{d y}{d x}) + 2 y = 10 x^{2}

\frac{\partial}{\partial y} (x^{2} + y^{2} - 5)

\int \frac{6}{5 ( 4 - 7 x ) ^{3}} d x

\int \frac{1}{1 + t ^{4}} d t

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