What is the integral from 1 to 2 of sqrt(4r^2+1)*r ?

The integral from 1 to 2 of sqrt(4r^2+1)*r is (17sqrt(17)-5sqrt(5))/(12)

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integral from 1 to 2 of sqrt(4r^2+1)*r

\int_{1}^{2} 4 r^{2} + 1 \cdot r d r

\frac{17 17 - 5 5}{12}

Decimal Notation

4.90937 \dots

Solution steps

\int_{1}^{2} 4 r^{2} + 1 r d r

Apply u-substitution

: \int_{5}^{17} \frac{u}{8} d u

= \int_{5}^{17} \frac{u}{8} d u

Take the constant out:

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

= \frac{1}{8} \cdot \int_{5}^{17} u d u

Apply the Power Rule

: [\frac{2}{3} u^{\frac{3}{2}}]_{5}^{17}

= \frac{1}{8} [\frac{2}{3} u^{\frac{3}{2}}]_{5}^{17}

Compute the boundaries:

\frac{34 17 - 10 5}{3}

= \frac{1}{8} \cdot \frac{34 17 - 10 5}{3}

Simplify

= \frac{17 17 - 5 5}{12}

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What is the integral from 1 to 2 of sqrt(4r^2+1)*r ?
The integral from 1 to 2 of sqrt(4r^2+1)*r is (17sqrt(17)-5sqrt(5))/(12)