What is the integral of x(a^2-x^2)^{1/2} ?

The integral of x(a^2-x^2)^{1/2} is -1/3 (a^2-x^2)^{3/2}+C

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integral of x(a^2-x^2)^{1/2}

integral

x (a^{2} - x^{2})^{\frac{1}{2}}

- \frac{1}{3} (a^{2} - x^{2})^{\frac{3}{2}} + C

Solution steps

\int x (a^{2} - x^{2})^{\frac{1}{2}} d x

Apply u-substitution

= \int - \frac{u ^{\frac{1}{2}}}{2} d u

Take the constant out:

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

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

Apply the Power Rule

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

Substitute back

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

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

Simplify

- \frac{1}{2} \cdot \frac{2}{3} (a^{2} - x^{2})^{\frac{3}{2}} : - \frac{1}{3} (a^{2} - x^{2})^{\frac{3}{2}}

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

Add a constant to the solution

= - \frac{1}{3} (a^{2} - x^{2})^{\frac{3}{2}} + C

What is the integral of x(a^2-x^2)^{1/2} ?
The integral of x(a^2-x^2)^{1/2} is -1/3 (a^2-x^2)^{3/2}+C