What is the integral of xsqrt(12-x^2) ?

The integral of xsqrt(12-x^2) is -1/3 (12-x^2)^{3/2}+C

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integral of xsqrt(12-x^2)

\int x 12 - x^{2} d x

- \frac{1}{3} (12 - x^{2})^{\frac{3}{2}} + C

Solution steps

\int x 12 - x^{2} d x

Apply u-substitution

: \int - \frac{u}{2} d u

= \int - \frac{u}{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 d u

Apply the Power Rule

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

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

Substitute back

u = 12 - x^{2}

= - \frac{1}{2} \cdot \frac{2}{3} (12 - x^{2})^{\frac{3}{2}}

Simplify

- \frac{1}{2} \cdot \frac{2}{3} (12 - x^{2})^{\frac{3}{2}} : - \frac{1}{3} (12 - x^{2})^{\frac{3}{2}}

= - \frac{1}{3} (12 - x^{2})^{\frac{3}{2}}

Add a constant to the solution

= - \frac{1}{3} (12 - x^{2})^{\frac{3}{2}} + C

y = 1 - x^{2}, y = 4 - 4 x

f (x) = - 3 x

\frac{\partial}{\partial x} ((x^{2} y^{2})^{\frac{1}{3}})

\frac{\partial}{\partial x} (8 y e^{5 x})

\int \frac{5 ^{- x}}{ln ( 5 )} d x

What is the integral of xsqrt(12-x^2) ?
The integral of xsqrt(12-x^2) is -1/3 (12-x^2)^{3/2}+C