What is the integral of xe^{\sqrt[3]{x}} ?

The integral of xe^{\sqrt[3]{x}} is

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integral of xe^{\sqrt[3]{x}}

Solution steps

Apply u-substitution

: \int 3 e^{u} u^{5} d u

= \int 3 e^{u} u^{5} d u

Take the constant out:

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

= 3 \cdot \int e^{u} u^{5} d u

Apply Integration By Parts

: u^{5} e^{u} - \int 5 u^{4} e^{u} d u

= 3 (u^{5} e^{u} - \int 5 u^{4} e^{u} d u)

\int 5 u^{4} e^{u} d u = 5 (u^{4} e^{u} - 4 (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u}))))

= 3 (u^{5} e^{u} - 5 (u^{4} e^{u} - 4 (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))))

Substitute back

Simplify

Add a constant to the solution

\int 1 t^{3} t + \frac{1}{t t} d t

\int 15 ln (6 x + 10) d x

\int 6 tan (\frac{x}{3}) d x

\int 4 x^{2} (2 x^{3} - 5)^{6} d x