What is the integral of (x-sqrt(25-x^2)) ?

The integral of (x-sqrt(25-x^2)) is (x^2)/2-25/2 (arcsin(1/5 x)+1/2 sin(2arcsin(1/5 x)))+C

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integral of (x-sqrt(25-x^2))

\int (x - 25 - x^{2}) d x

\frac{x ^{2}}{2} - \frac{25}{2} (arcsin (\frac{1}{5} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{5} x))) + C

Solution steps

\int x - 25 - x^{2} d x

Apply the Sum Rule:

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

= \int x d x - \int 25 - x^{2} d x

\int x d x = \frac{x ^{2}}{2}

\int 25 - x^{2} d x = \frac{25}{2} (arcsin (\frac{1}{5} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{5} x)))

= \frac{x ^{2}}{2} - \frac{25}{2} (arcsin (\frac{1}{5} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{5} x)))

Add a constant to the solution

= \frac{x ^{2}}{2} - \frac{25}{2} (arcsin (\frac{1}{5} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{5} x))) + C

What is the integral of (x-sqrt(25-x^2)) ?
The integral of (x-sqrt(25-x^2)) is (x^2)/2-25/2 (arcsin(1/5 x)+1/2 sin(2arcsin(1/5 x)))+C