What is the integral from 0 to 1 of x(sqrt(x)-x^3) ?

The integral from 0 to 1 of x(sqrt(x)-x^3) is 1/5

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integral from 0 to 1 of x(sqrt(x)-x^3)

\int_{0}^{1} x (x - x^{3}) d x

\frac{1}{5}

Decimal

0.2

Solution steps

\int_{0}^{1} x (x - x^{3}) d x

Apply Integration By Parts

= [x (\frac{2}{3} x^{\frac{3}{2}} - \frac{x ^{4}}{4}) - \int \frac{2}{3} x^{\frac{3}{2}} - \frac{x ^{4}}{4} d x]_{0}^{1}

\int \frac{2}{3} x^{\frac{3}{2}} - \frac{x ^{4}}{4} d x = \frac{4}{15} x^{\frac{5}{2}} - \frac{x ^{5}}{20}

= [x (\frac{2}{3} x^{\frac{3}{2}} - \frac{x ^{4}}{4}) - (\frac{4}{15} x^{\frac{5}{2}} - \frac{x ^{5}}{20})]_{0}^{1}

Simplify

[x (\frac{2}{3} x^{\frac{3}{2}} - \frac{x ^{4}}{4}) - (\frac{4}{15} x^{\frac{5}{2}} - \frac{x ^{5}}{20})]_{0}^{1} : [\frac{1}{5} (- x^{5} + 2 x^{\frac{5}{2}})]_{0}^{1}

= [\frac{1}{5} (- x^{5} + 2 x^{\frac{5}{2}})]_{0}^{1}

Compute the boundaries:

\frac{1}{5}

= \frac{1}{5}

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What is the integral from 0 to 1 of x(sqrt(x)-x^3) ?
The integral from 0 to 1 of x(sqrt(x)-x^3) is 1/5