What is the integral of xsqrt(64-16x^2) ?

The integral of xsqrt(64-16x^2) is -4/3 (-x^2+4)^{3/2}+C

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integral of xsqrt(64-16x^2)

\int x 64 - 16 x^{2} d x

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

Solution steps

\int x 64 - 16 x^{2} d x

Apply u-substitution

= \int - \frac{u}{32} d u

Take the constant out:

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

= - \frac{1}{32} \cdot \int u d u

Apply the Power Rule

= - \frac{1}{32} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 64 - 16 x^{2}

= - \frac{1}{32} \cdot \frac{2}{3} (64 - 16 x^{2})^{\frac{3}{2}}

Simplify

- \frac{1}{32} \cdot \frac{2}{3} (64 - 16 x^{2})^{\frac{3}{2}} : - \frac{4}{3} (- x^{2} + 4)^{\frac{3}{2}}

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

Add a constant to the solution

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

What is the integral of xsqrt(64-16x^2) ?
The integral of xsqrt(64-16x^2) is -4/3 (-x^2+4)^{3/2}+C