What is the integral of (x^2+x)^6(2x+1) ?

The integral of (x^2+x)^6(2x+1) is (x^{14})/7+x^{13}+3x^{12}+5x^{11}+5x^{10}+3x^9+x^8+(x^7)/7+C

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integral of (x^2+x)^6(2x+1)

\int (x^{2} + x)^{6} (2 x + 1) d x

\frac{x ^{14}}{7} + x^{13} + 3 x^{12} + 5 x^{11} + 5 x^{10} + 3 x^{9} + x^{8} + \frac{x ^{7}}{7} + C

Solution steps

\int (x^{2} + x)^{6} (2 x + 1) d x

Expand

(x^{2} + x)^{6} (2 x + 1) : 2 x^{13} + 13 x^{12} + 36 x^{11} + 55 x^{10} + 50 x^{9} + 27 x^{8} + 8 x^{7} + x^{6}

= \int 2 x^{13} + 13 x^{12} + 36 x^{11} + 55 x^{10} + 50 x^{9} + 27 x^{8} + 8 x^{7} + x^{6} 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 2 x^{13} d x + \int 13 x^{12} d x + \int 36 x^{11} d x + \int 55 x^{10} d x + \int 50 x^{9} d x + \int 27 x^{8} d x + \int 8 x^{7} d x + \int x^{6} d x

\int 2 x^{13} d x = \frac{x ^{14}}{7}

\int 13 x^{12} d x = x^{13}

\int 36 x^{11} d x = 3 x^{12}

\int 55 x^{10} d x = 5 x^{11}

\int 50 x^{9} d x = 5 x^{10}

\int 27 x^{8} d x = 3 x^{9}

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

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

= \frac{x ^{14}}{7} + x^{13} + 3 x^{12} + 5 x^{11} + 5 x^{10} + 3 x^{9} + x^{8} + \frac{x ^{7}}{7}

Add a constant to the solution

= \frac{x ^{14}}{7} + x^{13} + 3 x^{12} + 5 x^{11} + 5 x^{10} + 3 x^{9} + x^{8} + \frac{x ^{7}}{7} + C

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What is the integral of (x^2+x)^6(2x+1) ?
The integral of (x^2+x)^6(2x+1) is (x^{14})/7+x^{13}+3x^{12}+5x^{11}+5x^{10}+3x^9+x^8+(x^7)/7+C