What is the integral from 0 to 1 of x(3/2 x^2+1/2) ?

The integral from 0 to 1 of x(3/2 x^2+1/2) is 5/8

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integral from 0 to 1 of x(3/2 x^2+1/2)

\int_{0}^{1} x (\frac{3}{2} x^{2} + \frac{1}{2}) d x

\frac{5}{8}

Decimal

0.625

Solution steps

\int_{0}^{1} x (\frac{3}{2} x^{2} + \frac{1}{2}) d x

Multiply by the conjugate of

\frac{3}{2} x^{2} + \frac{1}{2} : \frac{\frac{9}{4} x ^{4} - \frac{1}{4}}{\frac{3}{2} x ^{2} - \frac{1}{2}}

= \int_{0}^{1} x \frac{\frac{9}{4} x ^{4} - \frac{1}{4}}{\frac{3}{2} x ^{2} - \frac{1}{2}} d x

Find undefined (singularity) points:

x = \frac{1}{3}

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_{0}^{\frac{1}{3}} x \frac{\frac{9}{4} x ^{4} - \frac{1}{4}}{\frac{3}{2} x ^{2} - \frac{1}{2}} d x + \int_{\frac{1}{3}}^{1} x \frac{\frac{9}{4} x ^{4} - \frac{1}{4}}{\frac{3}{2} x ^{2} - \frac{1}{2}} d x

\int_{0}^{\frac{1}{3}} x \frac{\frac{9}{4} x ^{4} - \frac{1}{4}}{\frac{3}{2} x ^{2} - \frac{1}{2}} d x = \frac{1}{8}

\int_{\frac{1}{3}}^{1} x \frac{\frac{9}{4} x ^{4} - \frac{1}{4}}{\frac{3}{2} x ^{2} - \frac{1}{2}} d x = \frac{1}{2}

= \frac{1}{8} + \frac{1}{2}

Simplify

\frac{1}{8} + \frac{1}{2} : \frac{5}{8}

= \frac{5}{8}

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What is the integral from 0 to 1 of x(3/2 x^2+1/2) ?
The integral from 0 to 1 of x(3/2 x^2+1/2) is 5/8