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

The integral from 1 to 4 of (1+x)/(sqrt(x)) is 20/3

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

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

\frac{20}{3}

Decimal

6.66666 \dots

Solution steps

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

Expand

\frac{1 + x}{x} : \frac{1}{x} + x

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

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

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

= 2 + \frac{14}{3}

Simplify

2 + \frac{14}{3} : \frac{20}{3}

= \frac{20}{3}

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What is the integral from 1 to 4 of (1+x)/(sqrt(x)) ?
The integral from 1 to 4 of (1+x)/(sqrt(x)) is 20/3