What is the integral of-2xsqrt(1-x^2) ?

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

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integral of-2xsqrt(1-x^2)

\int - 2 x 1 - x^{2} d x

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

Solution steps

\int - 2 x 1 - 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 1 - x^{2} d x

Apply u-substitution

: \int - u^{2} d u

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

Take the constant out:

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

= - 2 (- \int u^{2} d u)

Apply the Power Rule

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

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

Substitute back

u = 1 - x^{2}

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

Simplify

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

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

Add a constant to the solution

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

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x \to 8 - lim (\frac{4}{x - 8})

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

\int e^{x} (cos (x) + sin (x)) d x

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