What is the integral from 0 to 5 of x/(sqrt(x+4)) ?

The integral from 0 to 5 of x/(sqrt(x+4)) is 14/3

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integral from 0 to 5 of x/(sqrt(x+4))

\int_{0}^{5} \frac{x}{x + 4} d x

\frac{14}{3}

Decimal

4.66666 \dots

Solution steps

\int_{0}^{5} \frac{x}{x + 4} d x

Apply u-substitution

: \int_{2}^{3} 2 (u^{2} - 4) d u

= \int_{2}^{3} 2 (u^{2} - 4) 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} - 4 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} 4 d u)

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

\int_{2}^{3} 4 d u = 4

= 2 (\frac{19}{3} - 4)

Simplify

2 (\frac{19}{3} - 4) : \frac{14}{3}

= \frac{14}{3}

\int \frac{1}{1 + t ^{3}} d t

\int \frac{1}{7 ^{x}} d x

f (x) = - x^{2} + 7 x

\int_{- 1}^{1} x^{3} - 6 x d x

x^{4} + 5 x^{2} - x

What is the integral from 0 to 5 of x/(sqrt(x+4)) ?
The integral from 0 to 5 of x/(sqrt(x+4)) is 14/3