What is the integral from 0 to 1 of x^3sqrt(7-5x^4) ?

The integral from 0 to 1 of x^3sqrt(7-5x^4) is (7sqrt(7)-2sqrt(2))/(30)

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integral from 0 to 1 of x^3sqrt(7-5x^4)

\int_{0}^{1} x^{3} 7 - 5 x^{4} d x

\frac{7 7 - 2 2}{30}

Decimal

0.52306 \dots

Solution steps

\int_{0}^{1} x^{3} 7 - 5 x^{4} d x

Apply u-substitution

: \int_{7}^{2} - \frac{u}{20} d u

= \int_{7}^{2} - \frac{u}{20} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{2}^{7} - \frac{u}{20} d u

Take the constant out:

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

= - (- \frac{1}{20} \cdot \int_{2}^{7} u d u)

Apply the Power Rule

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

= - (- \frac{1}{20} [\frac{2}{3} u^{\frac{3}{2}}]_{2}^{7})

Simplify

= \frac{1}{20} [\frac{2}{3} u^{\frac{3}{2}}]_{2}^{7}

Compute the boundaries:

\frac{14 7 - 4 2}{3}

= \frac{1}{20} \cdot \frac{14 7 - 4 2}{3}

Simplify

= \frac{7 7 - 2 2}{30}

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What is the integral from 0 to 1 of x^3sqrt(7-5x^4) ?
The integral from 0 to 1 of x^3sqrt(7-5x^4) is (7sqrt(7)-2sqrt(2))/(30)