What is the integral from 0 to a^{1/3 of}x^5sqrt(x^6+a^2) ?

The integral from 0 to a^{1/3 of}x^5sqrt(x^6+a^2) is ((2sqrt(2)-1)a^3)/9

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integral from 0 to a^{1/3 of}x^5sqrt(x^6+a^2)

\int_{0}^{a^{\frac{1}{3}}} x^{5} x^{6} + a^{2} d x

\frac{( 2 2 - 1 ) a ^{3}}{9}

Solution steps

\int_{0}^{a^{\frac{1}{3}}} x^{5} x^{6} + a^{2} d x

Apply u-substitution

: \int_{a^{2}}^{2 a^{2}} \frac{u}{6} d u

= \int_{a^{2}}^{2 a^{2}} \frac{u}{6} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int_{a^{2}}^{2 a^{2}} u d u

Apply the Power Rule

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

= \frac{1}{6} [\frac{2}{3} u^{\frac{3}{2}}]_{a^{2}}^{2 a^{2}}

Compute the boundaries:

\frac{4 2 - 2}{3} a^{3}

= \frac{1}{6} \cdot \frac{4 2 - 2}{3} a^{3}

Simplify

= \frac{( 2 2 - 1 ) a ^{3}}{9}

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What is the integral from 0 to a^{1/3 of}x^5sqrt(x^6+a^2) ?
The integral from 0 to a^{1/3 of}x^5sqrt(x^6+a^2) is ((2sqrt(2)-1)a^3)/9