What is the integral of x/(\sqrt[3]{x+8)} ?

The integral of x/(\sqrt[3]{x+8)} is 3/5 (x+8)^{5/3}-12(x+8)^{2/3}+C

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integral of x/(\sqrt[3]{x+8)}

\int \frac{x}{3 x + 8} d x

\frac{3}{5} (x + 8)^{\frac{5}{3}} - 12 (x + 8)^{\frac{2}{3}} + C

Solution steps

\int \frac{x}{3 x + 8} d x

Apply u-substitution

= \int \frac{u - 8}{3 u} d u

Expand

\frac{u - 8}{3 u} : u^{\frac{2}{3}} - \frac{8}{3 u}

= \int u^{\frac{2}{3}} - \frac{8}{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

= \int u^{\frac{2}{3}} d u - \int \frac{8}{3 u} d u

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

\int \frac{8}{3 u} d u = 12 u^{\frac{2}{3}}

= \frac{3}{5} u^{\frac{5}{3}} - 12 u^{\frac{2}{3}}

Substitute back

u = x + 8

= \frac{3}{5} (x + 8)^{\frac{5}{3}} - 12 (x + 8)^{\frac{2}{3}}

Add a constant to the solution

= \frac{3}{5} (x + 8)^{\frac{5}{3}} - 12 (x + 8)^{\frac{2}{3}} + C

l a pl a ce t r an s f or m e^{3 t} cos (6 t)

\int 3 a^{7} x^{6} d x

\int (x + 1) e^{- x} d x

\int x e^{0.2 x} d x

d er i v a t i v e f (x) = x^{3} + 3 x

What is the integral of x/(\sqrt[3]{x+8)} ?
The integral of x/(\sqrt[3]{x+8)} is 3/5 (x+8)^{5/3}-12(x+8)^{2/3}+C