What is the integral of x^5(x^6-10)^4 ?

The integral of x^5(x^6-10)^4 is 1/30 (x^{30}-50x^{24}+1000x^{18}-10000x^{12}+50000x^6)+C

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integral of x^5(x^6-10)^4

\int x^{5} (x^{6} - 10)^{4} d x

\frac{1}{30} (x^{30} - 50 x^{24} + 1000 x^{18} - 10000 x^{12} + 50000 x^{6}) + C

Solution steps

\int x^{5} (x^{6} - 10)^{4} d x

Apply u-substitution

: \int \frac{u ^{\frac{1}{5}} ( u ^{\frac{6}{5}} - 10 ) ^{4}}{5} d u

= \int \frac{u ^{\frac{1}{5}} ( u ^{\frac{6}{5}} - 10 ) ^{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}} - 10)^{4} d u

Expand

u^{\frac{1}{5}} (u^{\frac{6}{5}} - 10)^{4} : u^{5} - 40 u^{\frac{19}{5}} + 600 u^{\frac{13}{5}} - 4000 u^{\frac{7}{5}} + 10000 u^{\frac{1}{5}}

= \frac{1}{5} \cdot \int u^{5} - 40 u^{\frac{19}{5}} + 600 u^{\frac{13}{5}} - 4000 u^{\frac{7}{5}} + 10000 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 40 u^{\frac{19}{5}} d u + \int 600 u^{\frac{13}{5}} d u - \int 4000 u^{\frac{7}{5}} d u + \int 10000 u^{\frac{1}{5}} d u)

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

\int 40 u^{\frac{19}{5}} d u = \frac{25}{3} u^{\frac{24}{5}}

\int 600 u^{\frac{13}{5}} d u = \frac{500}{3} u^{\frac{18}{5}}

\int 4000 u^{\frac{7}{5}} d u = \frac{5000}{3} u^{\frac{12}{5}}

\int 10000 u^{\frac{1}{5}} d u = \frac{25000}{3} u^{\frac{6}{5}}

= \frac{1}{5} (\frac{u ^{6}}{6} - \frac{25}{3} u^{\frac{24}{5}} + \frac{500}{3} u^{\frac{18}{5}} - \frac{5000}{3} u^{\frac{12}{5}} + \frac{25000}{3} u^{\frac{6}{5}})

Substitute back

u = x^{5}

= \frac{1}{5} (\frac{( x ^{5} ) ^{6}}{6} - \frac{25}{3} (x^{5})^{\frac{24}{5}} + \frac{500}{3} (x^{5})^{\frac{18}{5}} - \frac{5000}{3} (x^{5})^{\frac{12}{5}} + \frac{25000}{3} (x^{5})^{\frac{6}{5}})

Simplify

\frac{1}{5} (\frac{( x ^{5} ) ^{6}}{6} - \frac{25}{3} (x^{5})^{\frac{24}{5}} + \frac{500}{3} (x^{5})^{\frac{18}{5}} - \frac{5000}{3} (x^{5})^{\frac{12}{5}} + \frac{25000}{3} (x^{5})^{\frac{6}{5}}) : \frac{1}{30} (x^{30} - 50 x^{24} + 1000 x^{18} - 10000 x^{12} + 50000 x^{6})

= \frac{1}{30} (x^{30} - 50 x^{24} + 1000 x^{18} - 10000 x^{12} + 50000 x^{6})

Add a constant to the solution

= \frac{1}{30} (x^{30} - 50 x^{24} + 1000 x^{18} - 10000 x^{12} + 50000 x^{6}) + C

(x^{2} + 4) d y = (2 x - 8 y) d x

n \to 6 lim ((- 1)^{n} 7 n)

x \to 3 lim (- x^{2} + 9 x - 8)

f (x) = d cos (t) + t^{2} sin (t)

\frac{d}{d t} (1 + 4 t)

What is the integral of x^5(x^6-10)^4 ?
The integral of x^5(x^6-10)^4 is 1/30 (x^{30}-50x^{24}+1000x^{18}-10000x^{12}+50000x^6)+C