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

The integral of sqrt(x)(x^3-2/(x^2)+6) is 4/(sqrt(x))+2/9 x^{9/2}+4x^{3/2}+C

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

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

\frac{4}{x} + \frac{2}{9} x^{\frac{9}{2}} + 4 x^{\frac{3}{2}} + C

Solution steps

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

Expand

x (x^{3} - \frac{2}{x ^{2}} + 6) : - \frac{2}{x ^{\frac{3}{2}}} + x^{\frac{7}{2}} + 6 x^{\frac{1}{2}}

= \int - \frac{2}{x ^{\frac{3}{2}}} + x^{\frac{7}{2}} + 6 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 ^{\frac{3}{2}}} d x + \int x^{\frac{7}{2}} d x + \int 6 x^{\frac{1}{2}} d x

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

\int x^{\frac{7}{2}} d x = \frac{2}{9} x^{\frac{9}{2}}

\int 6 x^{\frac{1}{2}} d x = 4 x^{\frac{3}{2}}

= - (- \frac{4}{x}) + \frac{2}{9} x^{\frac{9}{2}} + 4 x^{\frac{3}{2}}

Simplify

= \frac{4}{x} + \frac{2}{9} x^{\frac{9}{2}} + 4 x^{\frac{3}{2}}

Add a constant to the solution

= \frac{4}{x} + \frac{2}{9} x^{\frac{9}{2}} + 4 x^{\frac{3}{2}} + C

x \to 0 lim (x^{2} - \frac{cos ( x )}{10000})

\frac{\partial}{\partial x} (ln (x + y^{3}) - z)

\int \frac{8 x ^{7}}{( x ^{8} + 1 ) ^{2}} d x

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

(1 + 3 t)^{2}

What is the integral of sqrt(x)(x^3-2/(x^2)+6) ?
The integral of sqrt(x)(x^3-2/(x^2)+6) is 4/(sqrt(x))+2/9 x^{9/2}+4x^{3/2}+C