What is the integral from 1 to 4 of ((1+sqrt(x))^{1/2})/(sqrt(x)) ?

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

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integral from 1 to 4 of ((1+sqrt(x))^{1/2})/(sqrt(x))

\int_{1}^{4} \frac{( 1 + x ) ^{\frac{1}{2}}}{x} d x

2 (23 - \frac{4 2}{3})

Decimal

3.15696 \dots

Solution steps

\int_{1}^{4} \frac{( 1 + x ) ^{\frac{1}{2}}}{x} d x

Simplify

\frac{( 1 + x ) ^{\frac{1}{2}}}{x} : \frac{1 + x}{x}

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

Simplify

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

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

Apply u-substitution

: \int_{2}^{3} 2 u d u

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

Apply the Power Rule

: [\frac{2}{3} u^{\frac{3}{2}}]_{2}^{3}

= 2 [\frac{2}{3} u^{\frac{3}{2}}]_{2}^{3}

Compute the boundaries:

23 - \frac{4 2}{3}

= 2 (23 - \frac{4 2}{3})

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