What is the integral of (x+3)(x^2+6x+7)^6 ?

The integral of (x+3)(x^2+6x+7)^6 is

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integral of (x+3)(x^2+6x+7)^6

\int (x + 3) (x^{2} + 6 x + 7)^{6} d x

\frac{x ^{14}}{14} + 3 x^{13} + \frac{115 x ^{12}}{2} + 666 x^{11} + \frac{10407 x ^{10}}{2} + 28989 x^{9} + \frac{237371 x ^{8}}{2} + \frac{2538420 x ^{7}}{7} + \frac{1661597 x ^{6}}{2} + 1420461 x^{5} + \frac{3569601 x ^{4}}{2} + 1599066 x^{3} + \frac{1932805 x ^{2}}{2} + 352947 x + C

Solution steps

\int (x + 3) (x^{2} + 6 x + 7)^{6} d x

Expand

(x + 3) (x^{2} + 6 x + 7)^{6} : x^{13} + 39 x^{12} + 690 x^{11} + 7326 x^{10} + 52035 x^{9} + 260901 x^{8} + 949484 x^{7} + 2538420 x^{6} + 4984791 x^{5} + 7102305 x^{4} + 7139202 x^{3} + 4797198 x^{2} + 1932805 x + 352947

= \int x^{13} + 39 x^{12} + 690 x^{11} + 7326 x^{10} + 52035 x^{9} + 260901 x^{8} + 949484 x^{7} + 2538420 x^{6} + 4984791 x^{5} + 7102305 x^{4} + 7139202 x^{3} + 4797198 x^{2} + 1932805 x + 352947 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^{13} d x + \int 39 x^{12} d x + \int 690 x^{11} d x + \int 7326 x^{10} d x + \int 52035 x^{9} d x + \int 260901 x^{8} d x + \int 949484 x^{7} d x + \int 2538420 x^{6} d x + \int 4984791 x^{5} d x + \int 7102305 x^{4} d x + \int 7139202 x^{3} d x + \int 4797198 x^{2} d x + \int 1932805 x d x + \int 352947 d x

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

\int 39 x^{12} d x = 3 x^{13}

\int 690 x^{11} d x = \frac{115 x ^{12}}{2}

\int 7326 x^{10} d x = 666 x^{11}

\int 52035 x^{9} d x = \frac{10407 x ^{10}}{2}

\int 260901 x^{8} d x = 28989 x^{9}

\int 949484 x^{7} d x = \frac{237371 x ^{8}}{2}

\int 2538420 x^{6} d x = \frac{2538420 x ^{7}}{7}

\int 4984791 x^{5} d x = \frac{1661597 x ^{6}}{2}

\int 7102305 x^{4} d x = 1420461 x^{5}

\int 7139202 x^{3} d x = \frac{3569601 x ^{4}}{2}

\int 4797198 x^{2} d x = 1599066 x^{3}

\int 1932805 x d x = \frac{1932805 x ^{2}}{2}

\int 352947 d x = 352947 x

= \frac{x ^{14}}{14} + 3 x^{13} + \frac{115 x ^{12}}{2} + 666 x^{11} + \frac{10407 x ^{10}}{2} + 28989 x^{9} + \frac{237371 x ^{8}}{2} + \frac{2538420 x ^{7}}{7} + \frac{1661597 x ^{6}}{2} + 1420461 x^{5} + \frac{3569601 x ^{4}}{2} + 1599066 x^{3} + \frac{1932805 x ^{2}}{2} + 352947 x

Add a constant to the solution

= \frac{x ^{14}}{14} + 3 x^{13} + \frac{115 x ^{12}}{2} + 666 x^{11} + \frac{10407 x ^{10}}{2} + 28989 x^{9} + \frac{237371 x ^{8}}{2} + \frac{2538420 x ^{7}}{7} + \frac{1661597 x ^{6}}{2} + 1420461 x^{5} + \frac{3569601 x ^{4}}{2} + 1599066 x^{3} + \frac{1932805 x ^{2}}{2} + 352947 x + C

\int \frac{sin ( 9 ln ( t ) )}{t} d t

d er i v a t i v e y = cos ((π x)^{8})

t an g e n t y = 3 e^{x} cos (x), (0, 3)

4 y^{^{''}} + 12 y^{^{'}} + 9 y = 0

d er i v a t i v e f (x) = \frac{1}{2} x^{8}

What is the integral of (x+3)(x^2+6x+7)^6 ?
The integral of (x+3)(x^2+6x+7)^6 is