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Popular >

integral of x(1-5x^2)^7

Solution

\int x (1 - 5 x^{2})^{7} d x

Solution

\frac{x ^{2}}{2} - \frac{35 x ^{4}}{4} + \frac{175 x ^{6}}{2} - \frac{4375 x ^{8}}{8} + \frac{4375 x ^{10}}{2} - \frac{21875 x ^{12}}{4} + \frac{15625 x ^{14}}{2} - \frac{78125 x ^{16}}{16} + C

Solution steps

\int x (1 - 5 x^{2})^{7} d x

Expand

x (1 - 5 x^{2})^{7} : x - 35 x^{3} + 525 x^{5} - 4375 x^{7} + 21875 x^{9} - 65625 x^{11} + 109375 x^{13} - 78125 x^{15}

= \int x - 35 x^{3} + 525 x^{5} - 4375 x^{7} + 21875 x^{9} - 65625 x^{11} + 109375 x^{13} - 78125 x^{15} 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 d x - \int 35 x^{3} d x + \int 525 x^{5} d x - \int 4375 x^{7} d x + \int 21875 x^{9} d x - \int 65625 x^{11} d x + \int 109375 x^{13} d x - \int 78125 x^{15} d x

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

\int 35 x^{3} d x = \frac{35 x ^{4}}{4}

\int 525 x^{5} d x = \frac{175 x ^{6}}{2}

\int 4375 x^{7} d x = \frac{4375 x ^{8}}{8}

\int 21875 x^{9} d x = \frac{4375 x ^{10}}{2}

\int 65625 x^{11} d x = \frac{21875 x ^{12}}{4}

\int 109375 x^{13} d x = \frac{15625 x ^{14}}{2}

\int 78125 x^{15} d x = \frac{78125 x ^{16}}{16}

= \frac{x ^{2}}{2} - \frac{35 x ^{4}}{4} + \frac{175 x ^{6}}{2} - \frac{4375 x ^{8}}{8} + \frac{4375 x ^{10}}{2} - \frac{21875 x ^{12}}{4} + \frac{15625 x ^{14}}{2} - \frac{78125 x ^{16}}{16}

Add a constant to the solution

= \frac{x ^{2}}{2} - \frac{35 x ^{4}}{4} + \frac{175 x ^{6}}{2} - \frac{4375 x ^{8}}{8} + \frac{4375 x ^{10}}{2} - \frac{21875 x ^{12}}{4} + \frac{15625 x ^{14}}{2} - \frac{78125 x ^{16}}{16} + C

Graph

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Frequently Asked Questions (FAQ)

What is the integral of x(1-5x^2)^7 ?
The integral of x(1-5x^2)^7 is