What is the integral from-1 to 1 of x(x^2+5x)^3 ?

The integral from-1 to 1 of x(x^2+5x)^3 is 380/7

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integral from-1 to 1 of x(x^2+5x)^3

\int_{- 1}^{1} x (x^{2} + 5 x)^{3} d x

\frac{380}{7}

Decimal

54.28571 \dots

Solution steps

\int_{- 1}^{1} x (x^{2} + 5 x)^{3} d x

Multiply by the conjugate of

x^{2} + 5 x : \frac{x ^{4} - 25 x ^{2}}{x ^{2} - 5 x}

= \int_{- 1}^{1} x (\frac{x ^{4} - 25 x ^{2}}{x ^{2} - 5 x})^{3} d x

Find undefined (singularity) points:

x = 0

If exist

b, a < b < c, f (b) =

undefined

, \int_{a}^{c} f (x) d x = \int_{a}^{b} f (x) d x + \int_{b}^{c} f (x) d x

= \int_{- 1}^{0} x (\frac{x ^{4} - 25 x ^{2}}{x ^{2} - 5 x})^{3} d x + \int_{0}^{1} x (\frac{x ^{4} - 25 x ^{2}}{x ^{2} - 5 x})^{3} d x

\int_{- 1}^{0} x (\frac{x ^{4} - 25 x ^{2}}{x ^{2} - 5 x})^{3} d x = \frac{813}{56}

\int_{0}^{1} x (\frac{x ^{4} - 25 x ^{2}}{x ^{2} - 5 x})^{3} d x = \frac{2227}{56}

= \frac{813}{56} + \frac{2227}{56}

Simplify

\frac{813}{56} + \frac{2227}{56} : \frac{380}{7}

= \frac{380}{7}

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What is the integral from-1 to 1 of x(x^2+5x)^3 ?
The integral from-1 to 1 of x(x^2+5x)^3 is 380/7