What is the integral of x^{1/3}(x^{4/3}+9)^8 ?

The integral of x^{1/3}(x^{4/3}+9)^8 is

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integral of x^{1/3}(x^{4/3}+9)^8

\int x^{\frac{1}{3}} (x^{\frac{4}{3}} + 9)^{8} d x

\frac{x ^{12}}{12} + \frac{27}{4} x^{\frac{32}{3}} + 243 x^{\frac{28}{3}} + 5103 x^{8} + \frac{137781}{2} x^{\frac{20}{3}} + \frac{1240029}{2} x^{\frac{16}{3}} + 3720087 x^{4} + 14348907 x^{\frac{8}{3}} + \frac{129140163}{4} x^{\frac{4}{3}} + C

Solution steps

\int x^{\frac{1}{3}} (x^{\frac{4}{3}} + 9)^{8} d x

Expand

x^{\frac{1}{3}} (x^{\frac{4}{3}} + 9)^{8} : x^{11} + 72 x^{\frac{29}{3}} + 2268 x^{\frac{25}{3}} + 40824 x^{7} + 459270 x^{\frac{17}{3}} + 3306744 x^{\frac{13}{3}} + 14880348 x^{3} + 38263752 x^{\frac{5}{3}} + 43046721 x^{\frac{1}{3}}

= \int x^{11} + 72 x^{\frac{29}{3}} + 2268 x^{\frac{25}{3}} + 40824 x^{7} + 459270 x^{\frac{17}{3}} + 3306744 x^{\frac{13}{3}} + 14880348 x^{3} + 38263752 x^{\frac{5}{3}} + 43046721 x^{\frac{1}{3}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{11} d x + \int 72 x^{\frac{29}{3}} d x + \int 2268 x^{\frac{25}{3}} d x + \int 40824 x^{7} d x + \int 459270 x^{\frac{17}{3}} d x + \int 3306744 x^{\frac{13}{3}} d x + \int 14880348 x^{3} d x + \int 38263752 x^{\frac{5}{3}} d x + \int 43046721 x^{\frac{1}{3}} d x

\int x^{11} d x = \frac{x ^{12}}{12}

\int 72 x^{\frac{29}{3}} d x = \frac{27}{4} x^{\frac{32}{3}}

\int 2268 x^{\frac{25}{3}} d x = 243 x^{\frac{28}{3}}

\int 40824 x^{7} d x = 5103 x^{8}

\int 459270 x^{\frac{17}{3}} d x = \frac{137781}{2} x^{\frac{20}{3}}

\int 3306744 x^{\frac{13}{3}} d x = \frac{1240029}{2} x^{\frac{16}{3}}

\int 14880348 x^{3} d x = 3720087 x^{4}

\int 38263752 x^{\frac{5}{3}} d x = 14348907 x^{\frac{8}{3}}

\int 43046721 x^{\frac{1}{3}} d x = \frac{129140163}{4} x^{\frac{4}{3}}

= \frac{x ^{12}}{12} + \frac{27}{4} x^{\frac{32}{3}} + 243 x^{\frac{28}{3}} + 5103 x^{8} + \frac{137781}{2} x^{\frac{20}{3}} + \frac{1240029}{2} x^{\frac{16}{3}} + 3720087 x^{4} + 14348907 x^{\frac{8}{3}} + \frac{129140163}{4} x^{\frac{4}{3}}

Add a constant to the solution

= \frac{x ^{12}}{12} + \frac{27}{4} x^{\frac{32}{3}} + 243 x^{\frac{28}{3}} + 5103 x^{8} + \frac{137781}{2} x^{\frac{20}{3}} + \frac{1240029}{2} x^{\frac{16}{3}} + 3720087 x^{4} + 14348907 x^{\frac{8}{3}} + \frac{129140163}{4} x^{\frac{4}{3}} + C

\int sin^{2} (2 x) cos^{5} (2 x) d x

\frac{1}{s ^{2} + 1}

\frac{1}{3} cot (9 x)

x \to - 2 - lim (f (x))

\frac{d}{d x} ((2 x^{2} - 9)^{- 1})

What is the integral of x^{1/3}(x^{4/3}+9)^8 ?
The integral of x^{1/3}(x^{4/3}+9)^8 is