What is the integral of x^5\sqrt[3]{(1+x^3)^2} ?

The integral of x^5\sqrt[3]{(1+x^3)^2} is 1/40 (8x^3(x^3+1)^{5/3}-3(x^3+1)^{8/3})+C

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integral of x^5\sqrt[3]{(1+x^3)^2}

\frac{1}{40} (8 x^{3} (x^{3} + 1)^{\frac{5}{3}} - 3 (x^{3} + 1)^{\frac{8}{3}}) + C

Solution steps

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

Apply u-substitution

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

Take the constant out:

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

= \frac{1}{5} \cdot \int u^{\frac{1}{5}} (u^{\frac{3}{5}} + 1)^{\frac{2}{3}} d u

Apply u-substitution

= \frac{1}{5} \cdot \int \frac{5 ( v - 1 ) v ^{\frac{2}{3}}}{3} d v

Take the constant out:

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

= \frac{1}{5} \cdot \frac{5}{3} \cdot \int (v - 1) v^{\frac{2}{3}} d v

Apply Integration By Parts

= \frac{1}{5} \cdot \frac{5}{3} (\frac{3}{5} v^{\frac{5}{3}} (v - 1) - \int \frac{3}{5} v^{\frac{5}{3}} d v)

\int \frac{3}{5} v^{\frac{5}{3}} d v = \frac{9}{40} v^{\frac{8}{3}}

= \frac{1}{5} \cdot \frac{5}{3} (\frac{3}{5} v^{\frac{5}{3}} (v - 1) - \frac{9}{40} v^{\frac{8}{3}})

Substitute back

= \frac{1}{5} \cdot \frac{5}{3} (\frac{3}{5} ((x^{5})^{\frac{3}{5}} + 1)^{\frac{5}{3}} ((x^{5})^{\frac{3}{5}} + 1 - 1) - \frac{9}{40} ((x^{5})^{\frac{3}{5}} + 1)^{\frac{8}{3}})

Simplify

\frac{1}{5} \cdot \frac{5}{3} (\frac{3}{5} ((x^{5})^{\frac{3}{5}} + 1)^{\frac{5}{3}} ((x^{5})^{\frac{3}{5}} + 1 - 1) - \frac{9}{40} ((x^{5})^{\frac{3}{5}} + 1)^{\frac{8}{3}}) : \frac{1}{40} (8 x^{3} (x^{3} + 1)^{\frac{5}{3}} - 3 (x^{3} + 1)^{\frac{8}{3}})

= \frac{1}{40} (8 x^{3} (x^{3} + 1)^{\frac{5}{3}} - 3 (x^{3} + 1)^{\frac{8}{3}})

Add a constant to the solution

= \frac{1}{40} (8 x^{3} (x^{3} + 1)^{\frac{5}{3}} - 3 (x^{3} + 1)^{\frac{8}{3}}) + C

What is the integral of x^5\sqrt[3]{(1+x^3)^2} ?
The integral of x^5\sqrt[3]{(1+x^3)^2} is 1/40 (8x^3(x^3+1)^{5/3}-3(x^3+1)^{8/3})+C