What is the integral of x^5(x^6-7)^4 ?

The integral of x^5(x^6-7)^4 is 1/30 (x^{30}-35x^{24}+490x^{18}-3430x^{12}+12005x^6)+C

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integral of x^5(x^6-7)^4

\int x^{5} (x^{6} - 7)^{4} d x

\frac{1}{30} (x^{30} - 35 x^{24} + 490 x^{18} - 3430 x^{12} + 12005 x^{6}) + C

Solution steps

\int x^{5} (x^{6} - 7)^{4} d x

Apply u-substitution

: \int \frac{u ^{\frac{1}{5}} ( u ^{\frac{6}{5}} - 7 ) ^{4}}{5} d u

= \int \frac{u ^{\frac{1}{5}} ( u ^{\frac{6}{5}} - 7 ) ^{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}} - 7)^{4} d u

Expand

u^{\frac{1}{5}} (u^{\frac{6}{5}} - 7)^{4} : u^{5} - 28 u^{\frac{19}{5}} + 294 u^{\frac{13}{5}} - 1372 u^{\frac{7}{5}} + 2401 u^{\frac{1}{5}}

= \frac{1}{5} \cdot \int u^{5} - 28 u^{\frac{19}{5}} + 294 u^{\frac{13}{5}} - 1372 u^{\frac{7}{5}} + 2401 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 28 u^{\frac{19}{5}} d u + \int 294 u^{\frac{13}{5}} d u - \int 1372 u^{\frac{7}{5}} d u + \int 2401 u^{\frac{1}{5}} d u)

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

\int 28 u^{\frac{19}{5}} d u = \frac{35}{6} u^{\frac{24}{5}}

\int 294 u^{\frac{13}{5}} d u = \frac{245}{3} u^{\frac{18}{5}}

\int 1372 u^{\frac{7}{5}} d u = \frac{1715}{3} u^{\frac{12}{5}}

\int 2401 u^{\frac{1}{5}} d u = \frac{12005}{6} u^{\frac{6}{5}}

= \frac{1}{5} (\frac{u ^{6}}{6} - \frac{35}{6} u^{\frac{24}{5}} + \frac{245}{3} u^{\frac{18}{5}} - \frac{1715}{3} u^{\frac{12}{5}} + \frac{12005}{6} u^{\frac{6}{5}})

Substitute back

u = x^{5}

= \frac{1}{5} (\frac{( x ^{5} ) ^{6}}{6} - \frac{35}{6} (x^{5})^{\frac{24}{5}} + \frac{245}{3} (x^{5})^{\frac{18}{5}} - \frac{1715}{3} (x^{5})^{\frac{12}{5}} + \frac{12005}{6} (x^{5})^{\frac{6}{5}})

Simplify

\frac{1}{5} (\frac{( x ^{5} ) ^{6}}{6} - \frac{35}{6} (x^{5})^{\frac{24}{5}} + \frac{245}{3} (x^{5})^{\frac{18}{5}} - \frac{1715}{3} (x^{5})^{\frac{12}{5}} + \frac{12005}{6} (x^{5})^{\frac{6}{5}}) : \frac{1}{30} (x^{30} - 35 x^{24} + 490 x^{18} - 3430 x^{12} + 12005 x^{6})

= \frac{1}{30} (x^{30} - 35 x^{24} + 490 x^{18} - 3430 x^{12} + 12005 x^{6})

Add a constant to the solution

= \frac{1}{30} (x^{30} - 35 x^{24} + 490 x^{18} - 3430 x^{12} + 12005 x^{6}) + C

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What is the integral of x^5(x^6-7)^4 ?
The integral of x^5(x^6-7)^4 is 1/30 (x^{30}-35x^{24}+490x^{18}-3430x^{12}+12005x^6)+C