What is the integral from 0 to 2 of 15sqrt(4x+1) ?

The integral from 0 to 2 of 15sqrt(4x+1) is 65

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integral from 0 to 2 of 15sqrt(4x+1)

\int_{0}^{2} 15 4 x + 1 d x

65

Solution steps

\int_{0}^{2} 15 4 x + 1 d x

Take the constant out:

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

= 15 \cdot \int_{0}^{2} 4 x + 1 d x

Apply u-substitution

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

= 15 \cdot \int_{1}^{9} \frac{u}{4} d u

Take the constant out:

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

= 15 \cdot \frac{1}{4} \cdot \int_{1}^{9} u d u

Apply the Power Rule

: [\frac{2}{3} u^{\frac{3}{2}}]_{1}^{9}

= 15 \cdot \frac{1}{4} [\frac{2}{3} u^{\frac{3}{2}}]_{1}^{9}

Simplify

15 \cdot \frac{1}{4} [\frac{2}{3} u^{\frac{3}{2}}]_{1}^{9} : \frac{15}{4} [\frac{2}{3} u^{\frac{3}{2}}]_{1}^{9}

= \frac{15}{4} [\frac{2}{3} u^{\frac{3}{2}}]_{1}^{9}

Compute the boundaries:

\frac{52}{3}

= \frac{15}{4} \cdot \frac{52}{3}

Simplify

= 65

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What is the integral from 0 to 2 of 15sqrt(4x+1) ?
The integral from 0 to 2 of 15sqrt(4x+1) is 65