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

The integral of x^6(x^7-7)^4 is 1/35 x^7(x^{28}-35x^{21}+490x^{14}-3430x^7+12005)+C

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

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

\frac{1}{35} x^{7} (x^{28} - 35 x^{21} + 490 x^{14} - 3430 x^{7} + 12005) + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Expand

u^{\frac{1}{6}} (u^{\frac{7}{6}} - 7)^{4} : u^{\frac{29}{6}} - 28 u^{\frac{11}{3}} + 294 u^{\frac{5}{2}} - 1372 u^{\frac{4}{3}} + 2401 u^{\frac{1}{6}}

= \frac{1}{6} \cdot \int u^{\frac{29}{6}} - 28 u^{\frac{11}{3}} + 294 u^{\frac{5}{2}} - 1372 u^{\frac{4}{3}} + 2401 u^{\frac{1}{6}} 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}{6} (\int u^{\frac{29}{6}} d u - \int 28 u^{\frac{11}{3}} d u + \int 294 u^{\frac{5}{2}} d u - \int 1372 u^{\frac{4}{3}} d u + \int 2401 u^{\frac{1}{6}} d u)

\int u^{\frac{29}{6}} d u = \frac{6}{35} u^{\frac{35}{6}}

\int 28 u^{\frac{11}{3}} d u = 6 u^{\frac{14}{3}}

\int 294 u^{\frac{5}{2}} d u = 84 u^{\frac{7}{2}}

\int 1372 u^{\frac{4}{3}} d u = 588 u^{\frac{7}{3}}

\int 2401 u^{\frac{1}{6}} d u = 2058 u^{\frac{7}{6}}

= \frac{1}{6} (\frac{6}{35} u^{\frac{35}{6}} - 6 u^{\frac{14}{3}} + 84 u^{\frac{7}{2}} - 588 u^{\frac{7}{3}} + 2058 u^{\frac{7}{6}})

Substitute back

u = x^{6}

= \frac{1}{6} (\frac{6}{35} (x^{6})^{\frac{35}{6}} - 6 (x^{6})^{\frac{14}{3}} + 84 (x^{6})^{\frac{7}{2}} - 588 (x^{6})^{\frac{7}{3}} + 2058 (x^{6})^{\frac{7}{6}})

Simplify

\frac{1}{6} (\frac{6}{35} (x^{6})^{\frac{35}{6}} - 6 (x^{6})^{\frac{14}{3}} + 84 (x^{6})^{\frac{7}{2}} - 588 (x^{6})^{\frac{7}{3}} + 2058 (x^{6})^{\frac{7}{6}}) : \frac{1}{35} x^{7} (x^{28} - 35 x^{21} + 490 x^{14} - 3430 x^{7} + 12005)

= \frac{1}{35} x^{7} (x^{28} - 35 x^{21} + 490 x^{14} - 3430 x^{7} + 12005)

Add a constant to the solution

= \frac{1}{35} x^{7} (x^{28} - 35 x^{21} + 490 x^{14} - 3430 x^{7} + 12005) + C

(x^{2} + 2 y^{2}) = x y y^{^{'}}, y (- 1) = 1

\int z d z

\int_{3}^{\infty} \frac{4}{x ^{2} - x} d x

\int 8 x ln (2 x) d x

x \to 1 lim (x^{2} + x - 3)

What is the integral of x^6(x^7-7)^4 ?
The integral of x^6(x^7-7)^4 is 1/35 x^7(x^{28}-35x^{21}+490x^{14}-3430x^7+12005)+C