What is the integral of xsqrt(16-4x^2) ?

The integral of xsqrt(16-4x^2) is -2/3 (4-x^2)^{3/2}+C

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integral of xsqrt(16-4x^2)

\int x 16 - 4 x^{2} d x

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

Solution steps

\int x 16 - 4 x^{2} d x

Apply u-substitution

: \int - \frac{u}{8} d u

= \int - \frac{u}{8} d u

Take the constant out:

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

= - \frac{1}{8} \cdot \int u d u

Apply the Power Rule

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

= - \frac{1}{8} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 16 - 4 x^{2}

= - \frac{1}{8} \cdot \frac{2}{3} (16 - 4 x^{2})^{\frac{3}{2}}

Simplify

- \frac{1}{8} \cdot \frac{2}{3} (16 - 4 x^{2})^{\frac{3}{2}} : - \frac{2}{3} (4 - x^{2})^{\frac{3}{2}}

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

Add a constant to the solution

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

\frac{d}{d x} (5 ln (12 x))

x \to 0 lim (\frac{x ^{3} - 1}{x})

x \to 3 lim (\frac{2 x}{x - 3})

\frac{d y}{d x} = e^{x + y - 1} - 1

x \to 3 lim (\frac{x + 5}{x - 3})

What is the integral of xsqrt(16-4x^2) ?
The integral of xsqrt(16-4x^2) is -2/3 (4-x^2)^{3/2}+C