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

The integral of x(x+7)^6 is (x^8)/8+6x^7+(245x^6)/2+1372x^5+(36015x^4)/4+33614x^3+(117649x^2)/2+C

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integral of x(x+7)^6

\int x (x + 7)^{6} d x

\frac{x ^{8}}{8} + 6 x^{7} + \frac{245 x ^{6}}{2} + 1372 x^{5} + \frac{36015 x ^{4}}{4} + 33614 x^{3} + \frac{117649 x ^{2}}{2} + C

Solution steps

\int x (x + 7)^{6} d x

Expand

x (x + 7)^{6} : x^{7} + 42 x^{6} + 735 x^{5} + 6860 x^{4} + 36015 x^{3} + 100842 x^{2} + 117649 x

= \int x^{7} + 42 x^{6} + 735 x^{5} + 6860 x^{4} + 36015 x^{3} + 100842 x^{2} + 117649 x d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{7} d x + \int 42 x^{6} d x + \int 735 x^{5} d x + \int 6860 x^{4} d x + \int 36015 x^{3} d x + \int 100842 x^{2} d x + \int 117649 x d x

\int x^{7} d x = \frac{x ^{8}}{8}

\int 42 x^{6} d x = 6 x^{7}

\int 735 x^{5} d x = \frac{245 x ^{6}}{2}

\int 6860 x^{4} d x = 1372 x^{5}

\int 36015 x^{3} d x = \frac{36015 x ^{4}}{4}

\int 100842 x^{2} d x = 33614 x^{3}

\int 117649 x d x = \frac{117649 x ^{2}}{2}

= \frac{x ^{8}}{8} + 6 x^{7} + \frac{245 x ^{6}}{2} + 1372 x^{5} + \frac{36015 x ^{4}}{4} + 33614 x^{3} + \frac{117649 x ^{2}}{2}

Add a constant to the solution

= \frac{x ^{8}}{8} + 6 x^{7} + \frac{245 x ^{6}}{2} + 1372 x^{5} + \frac{36015 x ^{4}}{4} + 33614 x^{3} + \frac{117649 x ^{2}}{2} + C

\frac{d}{d x} (\frac{x ^{2} - 5 x + 3}{x})

\frac{d}{d x} (4 x^{2} + 5 x + 3)

\int \frac{4 y}{x} d x

\frac{\partial}{\partial y} (x y e^{- 6 y})

\frac{d}{d x} (ln (\frac{e ^{x} + 1}{e ^{x} - 1}))

What is the integral of x(x+7)^6 ?
The integral of x(x+7)^6 is (x^8)/8+6x^7+(245x^6)/2+1372x^5+(36015x^4)/4+33614x^3+(117649x^2)/2+C