What is the integral of 10x(5x^2+5)^6 ?

The integral of 10x(5x^2+5)^6 is

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integral of 10x(5x^2+5)^6

\int 10 x (5 x^{2} + 5)^{6} d x

10 (\frac{15625 x ^{14}}{14} + \frac{15625 x ^{12}}{2} + \frac{46875 x ^{10}}{2} + \frac{78125 x ^{8}}{2} + \frac{78125 x ^{6}}{2} + \frac{46875 x ^{4}}{2} + \frac{15625 x ^{2}}{2}) + C

Solution steps

\int 10 x (5 x^{2} + 5)^{6} d x

Take the constant out:

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

= 10 \cdot \int x (5 x^{2} + 5)^{6} d x

Expand

x (5 x^{2} + 5)^{6} : 15625 x^{13} + 93750 x^{11} + 234375 x^{9} + 312500 x^{7} + 234375 x^{5} + 93750 x^{3} + 15625 x

= 10 \cdot \int 15625 x^{13} + 93750 x^{11} + 234375 x^{9} + 312500 x^{7} + 234375 x^{5} + 93750 x^{3} + 15625 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

= 10 (\int 15625 x^{13} d x + \int 93750 x^{11} d x + \int 234375 x^{9} d x + \int 312500 x^{7} d x + \int 234375 x^{5} d x + \int 93750 x^{3} d x + \int 15625 x d x)

\int 15625 x^{13} d x = \frac{15625 x ^{14}}{14}

\int 93750 x^{11} d x = \frac{15625 x ^{12}}{2}

\int 234375 x^{9} d x = \frac{46875 x ^{10}}{2}

\int 312500 x^{7} d x = \frac{78125 x ^{8}}{2}

\int 234375 x^{5} d x = \frac{78125 x ^{6}}{2}

\int 93750 x^{3} d x = \frac{46875 x ^{4}}{2}

\int 15625 x d x = \frac{15625 x ^{2}}{2}

= 10 (\frac{15625 x ^{14}}{14} + \frac{15625 x ^{12}}{2} + \frac{46875 x ^{10}}{2} + \frac{78125 x ^{8}}{2} + \frac{78125 x ^{6}}{2} + \frac{46875 x ^{4}}{2} + \frac{15625 x ^{2}}{2})

Add a constant to the solution

= 10 (\frac{15625 x ^{14}}{14} + \frac{15625 x ^{12}}{2} + \frac{46875 x ^{10}}{2} + \frac{78125 x ^{8}}{2} + \frac{78125 x ^{6}}{2} + \frac{46875 x ^{4}}{2} + \frac{15625 x ^{2}}{2}) + C

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