What is the integral of xsqrt(4-3x^2) ?

The integral of xsqrt(4-3x^2) is -1/9 (4-3x^2)^{3/2}+C

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integral of xsqrt(4-3x^2)

\int x 4 - 3 x^{2} d x

- \frac{1}{9} (4 - 3 x^{2})^{\frac{3}{2}} + C

Solution steps

\int x 4 - 3 x^{2} d x

Apply u-substitution

: \int - \frac{u}{6} d u

= \int - \frac{u}{6} d u

Take the constant out:

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

= - \frac{1}{6} \cdot \int u d u

Apply the Power Rule

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

= - \frac{1}{6} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 4 - 3 x^{2}

= - \frac{1}{6} \cdot \frac{2}{3} (4 - 3 x^{2})^{\frac{3}{2}}

Simplify

- \frac{1}{6} \cdot \frac{2}{3} (4 - 3 x^{2})^{\frac{3}{2}} : - \frac{1}{9} (4 - 3 x^{2})^{\frac{3}{2}}

= - \frac{1}{9} (4 - 3 x^{2})^{\frac{3}{2}}

Add a constant to the solution

= - \frac{1}{9} (4 - 3 x^{2})^{\frac{3}{2}} + C

\frac{\partial}{\partial x} (x e^{\frac{y}{z}})

\frac{\partial}{\partial x} (arcsin (\frac{x}{y ^{2}}))

\frac{d}{d x} (A x e^{- x})

y = e^{π}

x \to \frac{π}{2} lim ((sin (x) - cos (x))^{t a n (x)})

What is the integral of xsqrt(4-3x^2) ?
The integral of xsqrt(4-3x^2) is -1/9 (4-3x^2)^{3/2}+C