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

The integral of x(x+6)^6 is (x^8)/8+(36x^7)/7+90x^6+864x^5+4860x^4+15552x^3+23328x^2+C

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integral of x(x+6)^6

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

\frac{x ^{8}}{8} + \frac{36 x ^{7}}{7} + 90 x^{6} + 864 x^{5} + 4860 x^{4} + 15552 x^{3} + 23328 x^{2} + C

Solution steps

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

Expand

x (x + 6)^{6} : x^{7} + 36 x^{6} + 540 x^{5} + 4320 x^{4} + 19440 x^{3} + 46656 x^{2} + 46656 x

= \int x^{7} + 36 x^{6} + 540 x^{5} + 4320 x^{4} + 19440 x^{3} + 46656 x^{2} + 46656 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 36 x^{6} d x + \int 540 x^{5} d x + \int 4320 x^{4} d x + \int 19440 x^{3} d x + \int 46656 x^{2} d x + \int 46656 x d x

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

\int 36 x^{6} d x = \frac{36 x ^{7}}{7}

\int 540 x^{5} d x = 90 x^{6}

\int 4320 x^{4} d x = 864 x^{5}

\int 19440 x^{3} d x = 4860 x^{4}

\int 46656 x^{2} d x = 15552 x^{3}

\int 46656 x d x = 23328 x^{2}

= \frac{x ^{8}}{8} + \frac{36 x ^{7}}{7} + 90 x^{6} + 864 x^{5} + 4860 x^{4} + 15552 x^{3} + 23328 x^{2}

Add a constant to the solution

= \frac{x ^{8}}{8} + \frac{36 x ^{7}}{7} + 90 x^{6} + 864 x^{5} + 4860 x^{4} + 15552 x^{3} + 23328 x^{2} + C

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What is the integral of x(x+6)^6 ?
The integral of x(x+6)^6 is (x^8)/8+(36x^7)/7+90x^6+864x^5+4860x^4+15552x^3+23328x^2+C