What is the integral of x^5(x^6-6)^4 ?

The integral of x^5(x^6-6)^4 is (x^{30})/(30)-x^{24}+12x^{18}-72x^{12}+216x^6+C

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integral of x^5(x^6-6)^4

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

\frac{x ^{30}}{30} - x^{24} + 12 x^{18} - 72 x^{12} + 216 x^{6} + C

Solution steps

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

Apply u-substitution

: \int \frac{u ^{\frac{1}{5}} ( u ^{\frac{6}{5}} - 6 ) ^{4}}{5} d u

= \int \frac{u ^{\frac{1}{5}} ( u ^{\frac{6}{5}} - 6 ) ^{4}}{5} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int u^{\frac{1}{5}} (u^{\frac{6}{5}} - 6)^{4} d u

Expand

u^{\frac{1}{5}} (u^{\frac{6}{5}} - 6)^{4} : u^{5} - 24 u^{\frac{19}{5}} + 216 u^{\frac{13}{5}} - 864 u^{\frac{7}{5}} + 1296 u^{\frac{1}{5}}

= \frac{1}{5} \cdot \int u^{5} - 24 u^{\frac{19}{5}} + 216 u^{\frac{13}{5}} - 864 u^{\frac{7}{5}} + 1296 u^{\frac{1}{5}} 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}{5} (\int u^{5} d u - \int 24 u^{\frac{19}{5}} d u + \int 216 u^{\frac{13}{5}} d u - \int 864 u^{\frac{7}{5}} d u + \int 1296 u^{\frac{1}{5}} d u)

\int u^{5} d u = \frac{u ^{6}}{6}

\int 24 u^{\frac{19}{5}} d u = 5 u^{\frac{24}{5}}

\int 216 u^{\frac{13}{5}} d u = 60 u^{\frac{18}{5}}

\int 864 u^{\frac{7}{5}} d u = 360 u^{\frac{12}{5}}

\int 1296 u^{\frac{1}{5}} d u = 1080 u^{\frac{6}{5}}

= \frac{1}{5} (\frac{u ^{6}}{6} - 5 u^{\frac{24}{5}} + 60 u^{\frac{18}{5}} - 360 u^{\frac{12}{5}} + 1080 u^{\frac{6}{5}})

Substitute back

u = x^{5}

= \frac{1}{5} (\frac{( x ^{5} ) ^{6}}{6} - 5 (x^{5})^{\frac{24}{5}} + 60 (x^{5})^{\frac{18}{5}} - 360 (x^{5})^{\frac{12}{5}} + 1080 (x^{5})^{\frac{6}{5}})

Simplify

\frac{1}{5} (\frac{( x ^{5} ) ^{6}}{6} - 5 (x^{5})^{\frac{24}{5}} + 60 (x^{5})^{\frac{18}{5}} - 360 (x^{5})^{\frac{12}{5}} + 1080 (x^{5})^{\frac{6}{5}}) : \frac{x ^{30}}{30} - x^{24} + 12 x^{18} - 72 x^{12} + 216 x^{6}

= \frac{x ^{30}}{30} - x^{24} + 12 x^{18} - 72 x^{12} + 216 x^{6}

Add a constant to the solution

= \frac{x ^{30}}{30} - x^{24} + 12 x^{18} - 72 x^{12} + 216 x^{6} + C

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\int 53 - (3 x^{2} + 2 x + 20) d x

What is the integral of x^5(x^6-6)^4 ?
The integral of x^5(x^6-6)^4 is (x^{30})/(30)-x^{24}+12x^{18}-72x^{12}+216x^6+C