What is the integral of 4xsqrt(1-x^2) ?

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

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integral of 4xsqrt(1-x^2)

\int 4 x 1 - x^{2} d x

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

Solution steps

\int 4 x 1 - x^{2} d x

Take the constant out:

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

= 4 \cdot \int x 1 - x^{2} d x

Apply u-substitution

: \int - u^{2} d u

= 4 \cdot \int - u^{2} d u

Take the constant out:

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

= 4 (- \int u^{2} d u)

Apply the Power Rule

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

= 4 (- \frac{u ^{3}}{3})

Substitute back

u = 1 - x^{2}

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

Simplify

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

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

Add a constant to the solution

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

\frac{d}{d x} (\frac{24}{( 1 - x ) ^{4}})

\frac{d}{d x} (\frac{- 1}{x ^{3}})

\frac{4}{5 x + 1}, at x = - 1

x \to 3 lim (3 - x)

x \to \infty lim (8 x)

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