What is the integral from-1 to 4 of (x/(sqrt(5+x))) ?

The integral from-1 to 4 of (x/(sqrt(5+x))) is 8/3

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integral from-1 to 4 of (x/(sqrt(5+x)))

\int_{- 1}^{4} (\frac{x}{5 + x}) d x

\frac{8}{3}

Decimal

2.66666 \dots

Solution steps

\int_{- 1}^{4} \frac{x}{5 + x} d x

Apply u-substitution

= \int_{2}^{3} 2 (u^{2} - 5) d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int_{2}^{3} u^{2} - 5 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

= 2 (\int_{2}^{3} u^{2} d u - \int_{2}^{3} 5 d u)

\int_{2}^{3} u^{2} d u = \frac{19}{3}

\int_{2}^{3} 5 d u = 5

= 2 (\frac{19}{3} - 5)

Simplify

2 (\frac{19}{3} - 5) : \frac{8}{3}

= \frac{8}{3}

What is the integral from-1 to 4 of (x/(sqrt(5+x))) ?
The integral from-1 to 4 of (x/(sqrt(5+x))) is 8/3