What is the integral of (x^3-7)^{2/3}x^8 ?

The integral of (x^3-7)^{2/3}x^8 is 1/220 (20(x^3-7)^{11/3}+385(x^3-7)^{8/3}+2156(x^3-7)^{5/3})+C

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integral of (x^3-7)^{2/3}x^8

\int (x^{3} - 7)^{\frac{2}{3}} x^{8} d x

\frac{1}{220} (20 (x^{3} - 7)^{\frac{11}{3}} + 385 (x^{3} - 7)^{\frac{8}{3}} + 2156 (x^{3} - 7)^{\frac{5}{3}}) + C

Solution steps

\int (x^{3} - 7)^{\frac{2}{3}} x^{8} d x

Apply u-substitution

: \int \frac{u ^{\frac{2}{3}} ( u + 7 ) ^{2}}{3} d u

= \int \frac{u ^{\frac{2}{3}} ( u + 7 ) ^{2}}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int u^{\frac{2}{3}} (u + 7)^{2} d u

Expand

u^{\frac{2}{3}} (u + 7)^{2} : u^{\frac{8}{3}} + 14 u^{\frac{5}{3}} + 49 u^{\frac{2}{3}}

= \frac{1}{3} \cdot \int u^{\frac{8}{3}} + 14 u^{\frac{5}{3}} + 49 u^{\frac{2}{3}} 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

= \frac{1}{3} (\int u^{\frac{8}{3}} d u + \int 14 u^{\frac{5}{3}} d u + \int 49 u^{\frac{2}{3}} d u)

\int u^{\frac{8}{3}} d u = \frac{3}{11} u^{\frac{11}{3}}

\int 14 u^{\frac{5}{3}} d u = \frac{21}{4} u^{\frac{8}{3}}

\int 49 u^{\frac{2}{3}} d u = \frac{147}{5} u^{\frac{5}{3}}

= \frac{1}{3} (\frac{3}{11} u^{\frac{11}{3}} + \frac{21}{4} u^{\frac{8}{3}} + \frac{147}{5} u^{\frac{5}{3}})

Substitute back

u = x^{3} - 7

= \frac{1}{3} (\frac{3}{11} (x^{3} - 7)^{\frac{11}{3}} + \frac{21}{4} (x^{3} - 7)^{\frac{8}{3}} + \frac{147}{5} (x^{3} - 7)^{\frac{5}{3}})

Simplify

\frac{1}{3} (\frac{3}{11} (x^{3} - 7)^{\frac{11}{3}} + \frac{21}{4} (x^{3} - 7)^{\frac{8}{3}} + \frac{147}{5} (x^{3} - 7)^{\frac{5}{3}}) : \frac{1}{220} (20 (x^{3} - 7)^{\frac{11}{3}} + 385 (x^{3} - 7)^{\frac{8}{3}} + 2156 (x^{3} - 7)^{\frac{5}{3}})

= \frac{1}{220} (20 (x^{3} - 7)^{\frac{11}{3}} + 385 (x^{3} - 7)^{\frac{8}{3}} + 2156 (x^{3} - 7)^{\frac{5}{3}})

Add a constant to the solution

= \frac{1}{220} (20 (x^{3} - 7)^{\frac{11}{3}} + 385 (x^{3} - 7)^{\frac{8}{3}} + 2156 (x^{3} - 7)^{\frac{5}{3}}) + C

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What is the integral of (x^3-7)^{2/3}x^8 ?
The integral of (x^3-7)^{2/3}x^8 is 1/220 (20(x^3-7)^{11/3}+385(x^3-7)^{8/3}+2156(x^3-7)^{5/3})+C