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

The integral of sqrt(x^3)(x^3+3/x) is 2/11 x^{3/2}(x^4+11)+C

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

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

\frac{2}{11} x^{\frac{3}{2}} (x^{4} + 11) + C

Solution steps

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

Apply u-substitution

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

Expand

= \frac{1}{3} \cdot \int u^{\frac{5}{6}} + \frac{3}{u} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{3} (\int u^{\frac{5}{6}} d u + \int \frac{3}{u} d u)

\int u^{\frac{5}{6}} d u = \frac{6}{11} u^{\frac{11}{6}}

\int \frac{3}{u} d u = 6 u

= \frac{1}{3} (\frac{6}{11} u^{\frac{11}{6}} + 6 u)

Substitute back

u = x^{3}

= \frac{1}{3} (\frac{6}{11} (x^{3})^{\frac{11}{6}} + 6 x^{3})

Simplify

\frac{1}{3} (\frac{6}{11} (x^{3})^{\frac{11}{6}} + 6 x^{3}) : \frac{2}{11} x^{\frac{3}{2}} (x^{4} + 11)

= \frac{2}{11} x^{\frac{3}{2}} (x^{4} + 11)

Add a constant to the solution

= \frac{2}{11} x^{\frac{3}{2}} (x^{4} + 11) + C

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What is the integral of sqrt(x^3)(x^3+3/x) ?
The integral of sqrt(x^3)(x^3+3/x) is 2/11 x^{3/2}(x^4+11)+C