What is the integral from 0 to 4 of xsqrt(x+1) ?

The integral from 0 to 4 of xsqrt(x+1) is 10sqrt(5)-2/5-(10sqrt(5)-2)/3

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integral from 0 to 4 of xsqrt(x+1)

\int_{0}^{4} x x + 1 d x

105 - \frac{2}{5} - \frac{10 5 - 2}{3}

Decimal

15.17378 \dots

Solution steps

\int_{0}^{4} x x + 1 d x

Apply u-substitution

= \int_{1}^{5} (u - 1) u d u

Expand

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

= \int_{1}^{5} 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_{1}^{5} u^{\frac{3}{2}} d u - \int_{1}^{5} u d u

\int_{1}^{5} u^{\frac{3}{2}} d u = 105 - \frac{2}{5}

\int_{1}^{5} u d u = \frac{10 5 - 2}{3}

= 105 - \frac{2}{5} - \frac{10 5 - 2}{3}

What is the integral from 0 to 4 of xsqrt(x+1) ?
The integral from 0 to 4 of xsqrt(x+1) is 10sqrt(5)-2/5-(10sqrt(5)-2)/3