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

The integral of (x^2+3)/(\sqrt[3]{x+2)} is 3/8 (x+2)^{8/3}-12/5 (x+2)^{5/3}+21/2 (x+2)^{2/3}+C

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

\int \frac{x ^{2} + 3}{3 x + 2} d x

\frac{3}{8} (x + 2)^{\frac{8}{3}} - \frac{12}{5} (x + 2)^{\frac{5}{3}} + \frac{21}{2} (x + 2)^{\frac{2}{3}} + C

Solution steps

\int \frac{x ^{2} + 3}{3 x + 2} d x

Apply u-substitution

= \int \frac{u ^{2} - 4 u + 7}{3 u} d u

Expand

\frac{u ^{2} - 4 u + 7}{3 u} : u^{\frac{5}{3}} - 4 u^{\frac{2}{3}} + \frac{7}{3 u}

= \int u^{\frac{5}{3}} - 4 u^{\frac{2}{3}} + \frac{7}{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{5}{3}} d u - \int 4 u^{\frac{2}{3}} d u + \int \frac{7}{3 u} d u

\int u^{\frac{5}{3}} d u = \frac{3}{8} u^{\frac{8}{3}}

\int 4 u^{\frac{2}{3}} d u = \frac{12}{5} u^{\frac{5}{3}}

\int \frac{7}{3 u} d u = \frac{21}{2} u^{\frac{2}{3}}

= \frac{3}{8} u^{\frac{8}{3}} - \frac{12}{5} u^{\frac{5}{3}} + \frac{21}{2} u^{\frac{2}{3}}

Substitute back

u = x + 2

= \frac{3}{8} (x + 2)^{\frac{8}{3}} - \frac{12}{5} (x + 2)^{\frac{5}{3}} + \frac{21}{2} (x + 2)^{\frac{2}{3}}

Add a constant to the solution

= \frac{3}{8} (x + 2)^{\frac{8}{3}} - \frac{12}{5} (x + 2)^{\frac{5}{3}} + \frac{21}{2} (x + 2)^{\frac{2}{3}} + C

\frac{\partial}{\partial x} (10 (11 x - 3 y + 12)^{5})

x \to 1 lim (sin (x))

r \to 9 lim (\frac{4 r - 3}{11})

\int \frac{x ^{2}}{x ^{6} + 25} d x

x \to 0 lim (x^{2} cos (x^{5}))

What is the integral of (x^2+3)/(\sqrt[3]{x+2)} ?
The integral of (x^2+3)/(\sqrt[3]{x+2)} is 3/8 (x+2)^{8/3}-12/5 (x+2)^{5/3}+21/2 (x+2)^{2/3}+C