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

The integral of (2-x)sqrt(4-x^2) is

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integral of (2-x)sqrt(4-x^2)

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

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

Solution steps

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

Apply Integration By Parts

: (2 - x) (2 arcsin (\frac{1}{2} x) + sin (2 arcsin (\frac{1}{2} x))) - \int - 2 arcsin (\frac{1}{2} x) - sin (2 arcsin (\frac{1}{2} x)) d x

= (2 - x) (2 arcsin (\frac{1}{2} x) + sin (2 arcsin (\frac{1}{2} x))) - \int - 2 arcsin (\frac{1}{2} x) - sin (2 arcsin (\frac{1}{2} x)) d x

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

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

Simplify

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

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

Add a constant to the solution

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

\int \frac{ln ( x ^{2} )}{x ^{2}} d x

\int_{0}^{2} 9 x^{3} x^{2} + 4 d x

2 x^{2} - x - 3

\int \frac{y ^{2} - b ^{2}}{y ^{2}} d y

\int \frac{e ^{x}}{1 + e ^{x}} d x