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Popular >

integral of x(x-3)^7

Solution

\int x (x - 3)^{7} d x

Solution

\frac{1}{2} x^{2} (x - 3)^{7} - \frac{7 x ^{9}}{18} + \frac{63 x ^{8}}{8} - \frac{135 x ^{7}}{2} + 315 x^{6} - \frac{1701}{2} x^{5} + \frac{5103 x ^{4}}{4} - \frac{1701}{2} x^{3} + C

Solution steps

\int x (x - 3)^{7} d x

Apply Integration By Parts

: \frac{1}{2} x^{2} (x - 3)^{7} - \int \frac{7}{2} x^{2} (x - 3)^{6} d x

= \frac{1}{2} x^{2} (x - 3)^{7} - \int \frac{7}{2} x^{2} (x - 3)^{6} d x

\int \frac{7}{2} x^{2} (x - 3)^{6} d x = \frac{7 x ^{9}}{18} - \frac{63 x ^{8}}{8} + \frac{135 x ^{7}}{2} - 315 x^{6} + \frac{1701}{2} x^{5} - \frac{5103 x ^{4}}{4} + \frac{1701}{2} x^{3}

= \frac{1}{2} x^{2} (x - 3)^{7} - (\frac{7 x ^{9}}{18} - \frac{63 x ^{8}}{8} + \frac{135 x ^{7}}{2} - 315 x^{6} + \frac{1701}{2} x^{5} - \frac{5103 x ^{4}}{4} + \frac{1701}{2} x^{3})

Simplify

= \frac{1}{2} x^{2} (x - 3)^{7} - \frac{7 x ^{9}}{18} + \frac{63 x ^{8}}{8} - \frac{135 x ^{7}}{2} + 315 x^{6} - \frac{1701}{2} x^{5} + \frac{5103 x ^{4}}{4} - \frac{1701}{2} x^{3}

Add a constant to the solution

= \frac{1}{2} x^{2} (x - 3)^{7} - \frac{7 x ^{9}}{18} + \frac{63 x ^{8}}{8} - \frac{135 x ^{7}}{2} + 315 x^{6} - \frac{1701}{2} x^{5} + \frac{5103 x ^{4}}{4} - \frac{1701}{2} x^{3} + C

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Frequently Asked Questions (FAQ)

What is the integral of x(x-3)^7 ?
The integral of x(x-3)^7 is