What is the integral from 0 to 1 of 4xsqrt(2x^2+5) ?

The integral from 0 to 1 of 4xsqrt(2x^2+5) is (14sqrt(7)-10sqrt(5))/3

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integral from 0 to 1 of 4xsqrt(2x^2+5)

\int_{0}^{1} 4 x 2 x^{2} + 5 d x

\frac{14 7 - 10 5}{3}

Decimal

4.89327 \dots

Solution steps

\int_{0}^{1} 4 x 2 x^{2} + 5 d x

Take the constant out:

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

= 4 \cdot \int_{0}^{1} x 2 x^{2} + 5 d x

Apply u-substitution

= 4 \cdot \int_{5}^{7} \frac{u}{4} d u

Take the constant out:

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

= 4 \cdot \frac{1}{4} \cdot \int_{5}^{7} u d u

Apply the Power Rule

= 4 \cdot \frac{1}{4} [\frac{2}{3} u^{\frac{3}{2}}]_{5}^{7}

Simplify

4 \cdot \frac{1}{4} [\frac{2}{3} u^{\frac{3}{2}}]_{5}^{7} : [\frac{2}{3} u^{\frac{3}{2}}]_{5}^{7}

= [\frac{2}{3} u^{\frac{3}{2}}]_{5}^{7}

Compute the boundaries:

\frac{14 7 - 10 5}{3}

= \frac{14 7 - 10 5}{3}

What is the integral from 0 to 1 of 4xsqrt(2x^2+5) ?
The integral from 0 to 1 of 4xsqrt(2x^2+5) is (14sqrt(7)-10sqrt(5))/3