What is the integral of-2xsqrt(8-x^2) ?

The integral of-2xsqrt(8-x^2) is 2/3 (8-x^2)^{3/2}+C

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integral of-2xsqrt(8-x^2)

\int - 2 x 8 - x^{2} d x

\frac{2}{3} (8 - x^{2})^{\frac{3}{2}} + C

Solution steps

\int - 2 x 8 - x^{2} d x

Take the constant out:

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

= - 2 \cdot \int x 8 - x^{2} d x

Apply u-substitution

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

Take the constant out:

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

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

Apply the Power Rule

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

Substitute back

u = 8 - x^{2}

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

Simplify

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

= \frac{2}{3} (8 - x^{2})^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{3} (8 - x^{2})^{\frac{3}{2}} + C

x \to 0 + lim (\frac{x - ∣ x ∣}{x})

x \to \infty lim (ln (3 x) - ln (x + 7))

\frac{d}{d x} (2 x x^{2} + 1)

\int \frac{6 x + 7}{( 9 x ^{2} + 21 x ) ^{3}} d x

\frac{d}{d x} (\frac{x ^{2}}{x ^{3} + 8})

What is the integral of-2xsqrt(8-x^2) ?
The integral of-2xsqrt(8-x^2) is 2/3 (8-x^2)^{3/2}+C