What is the integral from 0 to 30 of x/(sqrt(4+2x)) ?

The integral from 0 to 30 of x/(sqrt(4+2x)) is 72

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integral from 0 to 30 of x/(sqrt(4+2x))

\int_{0}^{30} \frac{x}{4 + 2 x} d x

72

Solution steps

\int_{0}^{30} \frac{x}{4 + 2 x} d x

Apply u-substitution

: \int_{2}^{8} \frac{u ^{2} - 4}{2} d u

= \int_{2}^{8} \frac{u ^{2} - 4}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int_{2}^{8} 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

= \frac{1}{2} (\int_{2}^{8} u^{2} d u - \int_{2}^{8} 4 d u)

\int_{2}^{8} u^{2} d u = 168

\int_{2}^{8} 4 d u = 24

= \frac{1}{2} (168 - 24)

Simplify

\frac{1}{2} (168 - 24) : 72

= 72

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What is the integral from 0 to 30 of x/(sqrt(4+2x)) ?
The integral from 0 to 30 of x/(sqrt(4+2x)) is 72