What is the integral of x^4(x^5-5)^3 ?

The integral of x^4(x^5-5)^3 is (x^{20})/(20)-x^{15}+15/2 x^{10}-25x^5+C

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integral of x^4(x^5-5)^3

\int x^{4} (x^{5} - 5)^{3} d x

\frac{x ^{20}}{20} - x^{15} + \frac{15}{2} x^{10} - 25 x^{5} + C

Solution steps

\int x^{4} (x^{5} - 5)^{3} d x

Apply u-substitution

= \int \frac{u ^{\frac{1}{4}} ( u ^{\frac{5}{4}} - 5 ) ^{3}}{4} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int u^{\frac{1}{4}} (u^{\frac{5}{4}} - 5)^{3} d u

Expand

u^{\frac{1}{4}} (u^{\frac{5}{4}} - 5)^{3} : u^{4} - 15 u^{\frac{11}{4}} + 75 u^{\frac{3}{2}} - 125 u^{\frac{1}{4}}

= \frac{1}{4} \cdot \int u^{4} - 15 u^{\frac{11}{4}} + 75 u^{\frac{3}{2}} - 125 u^{\frac{1}{4}} 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}{4} (\int u^{4} d u - \int 15 u^{\frac{11}{4}} d u + \int 75 u^{\frac{3}{2}} d u - \int 125 u^{\frac{1}{4}} d u)

\int u^{4} d u = \frac{u ^{5}}{5}

\int 15 u^{\frac{11}{4}} d u = 4 u^{\frac{15}{4}}

\int 75 u^{\frac{3}{2}} d u = 30 u^{\frac{5}{2}}

\int 125 u^{\frac{1}{4}} d u = 100 u^{\frac{5}{4}}

= \frac{1}{4} (\frac{u ^{5}}{5} - 4 u^{\frac{15}{4}} + 30 u^{\frac{5}{2}} - 100 u^{\frac{5}{4}})

Substitute back

u = x^{4}

= \frac{1}{4} (\frac{( x ^{4} ) ^{5}}{5} - 4 (x^{4})^{\frac{15}{4}} + 30 (x^{4})^{\frac{5}{2}} - 100 (x^{4})^{\frac{5}{4}})

Simplify

\frac{1}{4} (\frac{( x ^{4} ) ^{5}}{5} - 4 (x^{4})^{\frac{15}{4}} + 30 (x^{4})^{\frac{5}{2}} - 100 (x^{4})^{\frac{5}{4}}) : \frac{x ^{20}}{20} - x^{15} + \frac{15}{2} x^{10} - 25 x^{5}

= \frac{x ^{20}}{20} - x^{15} + \frac{15}{2} x^{10} - 25 x^{5}

Add a constant to the solution

= \frac{x ^{20}}{20} - x^{15} + \frac{15}{2} x^{10} - 25 x^{5} + C

What is the integral of x^4(x^5-5)^3 ?
The integral of x^4(x^5-5)^3 is (x^{20})/(20)-x^{15}+15/2 x^{10}-25x^5+C