What is the integral from 1 to 2 of x^2sqrt(x-1) ?

The integral from 1 to 2 of x^2sqrt(x-1) is 184/105

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integral from 1 to 2 of x^2sqrt(x-1)

\int_{1}^{2} x^{2} x - 1 d x

\frac{184}{105}

Decimal

1.75238 \dots

Solution steps

\int_{1}^{2} x^{2} x - 1 d x

Apply u-substitution

: \int_{0}^{1} (u + 1)^{2} u d u

= \int_{0}^{1} (u + 1)^{2} u d u

Expand

(u + 1)^{2} u : u^{\frac{5}{2}} + 2 u^{\frac{3}{2}} + u

= \int_{0}^{1} u^{\frac{5}{2}} + 2 u^{\frac{3}{2}} + u d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{0}^{1} u^{\frac{5}{2}} d u + \int_{0}^{1} 2 u^{\frac{3}{2}} d u + \int_{0}^{1} u d u

\int_{0}^{1} u^{\frac{5}{2}} d u = \frac{2}{7}

\int_{0}^{1} 2 u^{\frac{3}{2}} d u = \frac{4}{5}

\int_{0}^{1} u d u = \frac{2}{3}

= \frac{2}{7} + \frac{4}{5} + \frac{2}{3}

Simplify

\frac{2}{7} + \frac{4}{5} + \frac{2}{3} : \frac{184}{105}

= \frac{184}{105}

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What is the integral from 1 to 2 of x^2sqrt(x-1) ?
The integral from 1 to 2 of x^2sqrt(x-1) is 184/105