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

The integral from 0 to a of xsqrt(x^2+a^2) is (2sqrt(2)a^3-a^3)/3

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integral from 0 to a of xsqrt(x^2+a^2)

\int_{0}^{a} x x^{2} + a^{2} d x

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

Solution steps

\int_{0}^{a} x x^{2} + a^{2} d x

Apply u-substitution

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

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

Apply the Power Rule

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

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

Simplify

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

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

Compute the boundaries:

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

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

Simplify

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

\frac{d}{d x} (sin (2) (x))

- \frac{y}{x + e ^{y}}

\int_{0}^{3} \frac{6 t}{( t - 4 ) ^{2}} d t

\int \frac{2 x e ^{x}}{( x + 1 ) ^{2}} d x

\frac{\partial}{\partial x} (y - x^{2} y e^{- y})

What is the integral from 0 to a of xsqrt(x^2+a^2) ?
The integral from 0 to a of xsqrt(x^2+a^2) is (2sqrt(2)a^3-a^3)/3