What is the integral from 1 to 2 of xsqrt(x+1) ?

The integral from 1 to 2 of xsqrt(x+1) is -2sqrt(3)+(-4sqrt(2)+54sqrt(3))/(15)

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integral from 1 to 2 of xsqrt(x+1)

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

- 23 + \frac{- 4 2 + 54 3}{15}

Decimal

2.39415 \dots

Solution steps

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

Apply u-substitution

: \int_{2}^{3} (u - 1) u d u

= \int_{2}^{3} (u - 1) u d u

Expand

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

= \int_{2}^{3} 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_{2}^{3} u^{\frac{3}{2}} d u - \int_{2}^{3} u d u

\int_{2}^{3} u^{\frac{3}{2}} d u = \frac{18 3 - 8 2}{5}

\int_{2}^{3} u d u = 23 - \frac{4 2}{3}

= \frac{18 3 - 8 2}{5} - (23 - \frac{4 2}{3})

Simplify

\frac{18 3 - 8 2}{5} - (23 - \frac{4 2}{3}) : - 23 + \frac{- 4 2 + 54 3}{15}

= - 23 + \frac{- 4 2 + 54 3}{15}

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What is the integral from 1 to 2 of xsqrt(x+1) ?
The integral from 1 to 2 of xsqrt(x+1) is -2sqrt(3)+(-4sqrt(2)+54sqrt(3))/(15)