What is the integral of 2/3 x(a^2+x^2)^{3/2} ?

The integral of 2/3 x(a^2+x^2)^{3/2} is 2/15 (a^2+x^2)^{5/2}+C

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integral of 2/3 x(a^2+x^2)^{3/2}

\int \frac{2}{3} x (a^{2} + x^{2})^{\frac{3}{2}} d x

\frac{2}{15} (a^{2} + x^{2})^{\frac{5}{2}} + C

Solution steps

\int \frac{2}{3} x (a^{2} + x^{2})^{\frac{3}{2}} d x

Take the constant out:

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

= \frac{2}{3} \cdot \int x (a^{2} + x^{2})^{\frac{3}{2}} d x

Apply u-substitution

: \int \frac{u ^{\frac{3}{2}}}{2} d u

= \frac{2}{3} \cdot \int \frac{u ^{\frac{3}{2}}}{2} d u

Take the constant out:

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

= \frac{2}{3} \cdot \frac{1}{2} \cdot \int u^{\frac{3}{2}} d u

Apply the Power Rule

: \frac{2}{5} u^{\frac{5}{2}}

= \frac{2}{3} \cdot \frac{1}{2} \cdot \frac{2}{5} u^{\frac{5}{2}}

Substitute back

u = a^{2} + x^{2}

= \frac{2}{3} \cdot \frac{1}{2} \cdot \frac{2}{5} (a^{2} + x^{2})^{\frac{5}{2}}

Simplify

\frac{2}{3} \cdot \frac{1}{2} \cdot \frac{2}{5} (a^{2} + x^{2})^{\frac{5}{2}} : \frac{2}{15} (a^{2} + x^{2})^{\frac{5}{2}}

= \frac{2}{15} (a^{2} + x^{2})^{\frac{5}{2}}

Add a constant to the solution

= \frac{2}{15} (a^{2} + x^{2})^{\frac{5}{2}} + C

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What is the integral of 2/3 x(a^2+x^2)^{3/2} ?
The integral of 2/3 x(a^2+x^2)^{3/2} is 2/15 (a^2+x^2)^{5/2}+C