What is the integral from 0 to x of sqrt(3t+5) ?

The integral from 0 to x of sqrt(3t+5) is (2(3x+5)^{3/2}-10sqrt(5))/9

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integral from 0 to x of sqrt(3t+5)

\int_{0}^{x} 3 t + 5 d t

\frac{2 ( 3 x + 5 ) ^{\frac{3}{2}} - 10 5}{9}

Solution steps

\int_{0}^{x} 3 t + 5 d t

Apply u-substitution

: \int_{5}^{3 x + 5} \frac{u}{3} d u

= \int_{5}^{3 x + 5} \frac{u}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int_{5}^{3 x + 5} u d u

Apply the Power Rule

: [\frac{2}{3} u^{\frac{3}{2}}]_{5}^{3 x + 5}

= \frac{1}{3} [\frac{2}{3} u^{\frac{3}{2}}]_{5}^{3 x + 5}

Compute the boundaries:

\frac{2}{3} (3 x + 5)^{\frac{3}{2}} - \frac{10 5}{3}

= \frac{1}{3} (\frac{2}{3} (3 x + 5)^{\frac{3}{2}} - \frac{10 5}{3})

Simplify

= \frac{2 ( 3 x + 5 ) ^{\frac{3}{2}} - 10 5}{9}

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What is the integral from 0 to x of sqrt(3t+5) ?
The integral from 0 to x of sqrt(3t+5) is (2(3x+5)^{3/2}-10sqrt(5))/9