What is the integral of 14x(7x^2+7)^7 ?

The integral of 14x(7x^2+7)^7 is

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integral of 14x(7x^2+7)^7

\int 14 x (7 x^{2} + 7)^{7} d x

14 (\frac{823543 x ^{16}}{16} + \frac{823543 x ^{14}}{2} + \frac{5764801 x ^{12}}{4} + \frac{5764801 x ^{10}}{2} + \frac{28824005 x ^{8}}{8} + \frac{5764801 x ^{6}}{2} + \frac{5764801 x ^{4}}{4} + \frac{823543 x ^{2}}{2}) + C

Solution steps

\int 14 x (7 x^{2} + 7)^{7} d x

Take the constant out:

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

= 14 \cdot \int x (7 x^{2} + 7)^{7} d x

Expand

x (7 x^{2} + 7)^{7} : 823543 x^{15} + 5764801 x^{13} + 17294403 x^{11} + 28824005 x^{9} + 28824005 x^{7} + 17294403 x^{5} + 5764801 x^{3} + 823543 x

= 14 \cdot \int 823543 x^{15} + 5764801 x^{13} + 17294403 x^{11} + 28824005 x^{9} + 28824005 x^{7} + 17294403 x^{5} + 5764801 x^{3} + 823543 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

= 14 (\int 823543 x^{15} d x + \int 5764801 x^{13} d x + \int 17294403 x^{11} d x + \int 28824005 x^{9} d x + \int 28824005 x^{7} d x + \int 17294403 x^{5} d x + \int 5764801 x^{3} d x + \int 823543 x d x)

\int 823543 x^{15} d x = \frac{823543 x ^{16}}{16}

\int 5764801 x^{13} d x = \frac{823543 x ^{14}}{2}

\int 17294403 x^{11} d x = \frac{5764801 x ^{12}}{4}

\int 28824005 x^{9} d x = \frac{5764801 x ^{10}}{2}

\int 28824005 x^{7} d x = \frac{28824005 x ^{8}}{8}

\int 17294403 x^{5} d x = \frac{5764801 x ^{6}}{2}

\int 5764801 x^{3} d x = \frac{5764801 x ^{4}}{4}

\int 823543 x d x = \frac{823543 x ^{2}}{2}

= 14 (\frac{823543 x ^{16}}{16} + \frac{823543 x ^{14}}{2} + \frac{5764801 x ^{12}}{4} + \frac{5764801 x ^{10}}{2} + \frac{28824005 x ^{8}}{8} + \frac{5764801 x ^{6}}{2} + \frac{5764801 x ^{4}}{4} + \frac{823543 x ^{2}}{2})

Add a constant to the solution

= 14 (\frac{823543 x ^{16}}{16} + \frac{823543 x ^{14}}{2} + \frac{5764801 x ^{12}}{4} + \frac{5764801 x ^{10}}{2} + \frac{28824005 x ^{8}}{8} + \frac{5764801 x ^{6}}{2} + \frac{5764801 x ^{4}}{4} + \frac{823543 x ^{2}}{2}) + C

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