What is the integral from 0 to 1 of sqrt(1+9/4 x) ?

The integral from 0 to 1 of sqrt(1+9/4 x) is (13sqrt(13)-8)/(27)

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integral from 0 to 1 of sqrt(1+9/4 x)

\int_{0}^{1} 1 + \frac{9}{4} x d x

\frac{13 13 - 8}{27}

Decimal

1.43970 \dots

Solution steps

\int_{0}^{1} 1 + \frac{9}{4} x d x

Apply u-substitution

: \int_{1}^{\frac{13}{4}} \frac{4 u}{9} d u

= \int_{1}^{\frac{13}{4}} \frac{4 u}{9} d u

Take the constant out:

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

= \frac{4}{9} \cdot \int_{1}^{\frac{13}{4}} u d u

Apply the Power Rule

: [\frac{2}{3} u^{\frac{3}{2}}]_{1}^{\frac{13}{4}}

= \frac{4}{9} [\frac{2}{3} u^{\frac{3}{2}}]_{1}^{\frac{13}{4}}

Compute the boundaries:

\frac{13 13}{12} - \frac{2}{3}

= \frac{4}{9} (\frac{13 13}{12} - \frac{2}{3})

Simplify

= \frac{13 13 - 8}{27}

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What is the integral from 0 to 1 of sqrt(1+9/4 x) ?
The integral from 0 to 1 of sqrt(1+9/4 x) is (13sqrt(13)-8)/(27)