What is the integral of x^5(x^3+1)^{1/2} ?

The integral of x^5(x^3+1)^{1/2} is 1/3 (2/5 (x^3+1)^{5/2}-2/3 (x^3+1)^{3/2})+C

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integral of x^5(x^3+1)^{1/2}

\int x^{5} (x^{3} + 1)^{\frac{1}{2}} d x

\frac{1}{3} (\frac{2}{5} (x^{3} + 1)^{\frac{5}{2}} - \frac{2}{3} (x^{3} + 1)^{\frac{3}{2}}) + C

Solution steps

\int x^{5} (x^{3} + 1)^{\frac{1}{2}} d x

Apply u-substitution

= \int \frac{( u - 1 ) u ^{\frac{1}{2}}}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int (u - 1) u^{\frac{1}{2}} d u

Expand

(u - 1) u^{\frac{1}{2}} : u^{\frac{3}{2}} - u^{\frac{1}{2}}

= \frac{1}{3} \cdot \int u^{\frac{3}{2}} - u^{\frac{1}{2}} 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{3}{2}} d u - \int u^{\frac{1}{2}} d u)

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

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

= \frac{1}{3} (\frac{2}{5} u^{\frac{5}{2}} - \frac{2}{3} u^{\frac{3}{2}})

Substitute back

u = x^{3} + 1

= \frac{1}{3} (\frac{2}{5} (x^{3} + 1)^{\frac{5}{2}} - \frac{2}{3} (x^{3} + 1)^{\frac{3}{2}})

Add a constant to the solution

= \frac{1}{3} (\frac{2}{5} (x^{3} + 1)^{\frac{5}{2}} - \frac{2}{3} (x^{3} + 1)^{\frac{3}{2}}) + C

What is the integral of x^5(x^3+1)^{1/2} ?
The integral of x^5(x^3+1)^{1/2} is 1/3 (2/5 (x^3+1)^{5/2}-2/3 (x^3+1)^{3/2})+C