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

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

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integral of xsqrt(4x-x^2)

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

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

Solution steps

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

Complete the square

4 x - x^{2} : - (x - 2)^{2} + 4

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

Apply u-substitution

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

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

Expand

(u + 2) - u^{2} + 4 : u - u^{2} + 4 + 2 - u^{2} + 4

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int u - u^{2} + 4 d u = - \frac{1}{3} (- u^{2} + 4)^{\frac{3}{2}}

\int 2 - u^{2} + 4 d u = 4 (arcsin (\frac{1}{2} u) + \frac{1}{2} sin (2 arcsin (\frac{1}{2} u)))

= - \frac{1}{3} (- u^{2} + 4)^{\frac{3}{2}} + 4 (arcsin (\frac{1}{2} u) + \frac{1}{2} sin (2 arcsin (\frac{1}{2} u)))

Substitute back

u = x - 2

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

Simplify

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

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

Add a constant to the solution

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

\frac{d}{d x} (\frac{2 e ^{l n^{2} (x)} ln ( x )}{x})

y^{^{''}} - 6 y^{^{'}} + 3 y = 0

x \to 3 lim (\frac{x ^{2} - 9}{x - 3})

f (x) = \frac{( x ^{2} - 2 x - 48 )}{x + 6}

2 x^{2} - 1

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