What is the integral from 1 to 4 of (2-x)/(sqrt(x)) ?

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

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integral from 1 to 4 of (2-x)/(sqrt(x))

\int_{1}^{4} \frac{2 - x}{x} d x

- \frac{2}{3}

Decimal

- 0.66666 \dots

Solution steps

\int_{1}^{4} \frac{2 - x}{x} d x

Expand

\frac{2 - x}{x} : \frac{2}{x} - x

= \int_{1}^{4} \frac{2}{x} - x d x

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}^{4} \frac{2}{x} d x - \int_{1}^{4} x d x

\int_{1}^{4} \frac{2}{x} d x = 4

\int_{1}^{4} x d x = \frac{14}{3}

= 4 - \frac{14}{3}

Simplify

4 - \frac{14}{3} : - \frac{2}{3}

= - \frac{2}{3}

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