What is the integral from 0 to k of x*sqrt(x^2+4) ?

The integral from 0 to k of x*sqrt(x^2+4) is ((k^2+4)^{3/2}-8)/3

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integral from 0 to k of x*sqrt(x^2+4)

\int_{0}^{k} x \cdot x^{2} + 4 d x

\frac{( k ^{2} + 4 ) ^{\frac{3}{2}} - 8}{3}

Solution steps

\int_{0}^{k} x x^{2} + 4 d x

Apply u-substitution

: \int_{2}^{k^{2} + 4} u^{2} d u

= \int_{2}^{k^{2} + 4} u^{2} d u

Apply the Power Rule

: [\frac{u ^{3}}{3}]_{2}^{k^{2} + 4}

= [\frac{u ^{3}}{3}]_{2}^{k^{2} + 4}

Compute the boundaries:

\frac{( k ^{2} + 4 ) ^{\frac{3}{2}}}{3} - \frac{8}{3}

= \frac{( k ^{2} + 4 ) ^{\frac{3}{2}}}{3} - \frac{8}{3}

Simplify

= \frac{( k ^{2} + 4 ) ^{\frac{3}{2}} - 8}{3}

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What is the integral from 0 to k of x*sqrt(x^2+4) ?
The integral from 0 to k of x*sqrt(x^2+4) is ((k^2+4)^{3/2}-8)/3