What is the integral of sqrt(5+4x-x^2)+4 ?

The integral of sqrt(5+4x-x^2)+4 is 9/2 (arcsin(1/3 (x-2))+1/2 sin(2arcsin(1/3 (x-2))))+4x+C

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integral of sqrt(5+4x-x^2)+4

\int 5 + 4 x - x^{2} + 4 d x

\frac{9}{2} (arcsin (\frac{1}{3} (x - 2)) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} (x - 2)))) + 4 x + C

Solution steps

\int 5 + 4 x - x^{2} + 4 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 5 + 4 x - x^{2} d x + \int 4 d x

\int 5 + 4 x - x^{2} d x = \frac{9}{2} (arcsin (\frac{1}{3} (x - 2)) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} (x - 2))))

\int 4 d x = 4 x

= \frac{9}{2} (arcsin (\frac{1}{3} (x - 2)) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} (x - 2)))) + 4 x

Add a constant to the solution

= \frac{9}{2} (arcsin (\frac{1}{3} (x - 2)) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} (x - 2)))) + 4 x + C

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What is the integral of sqrt(5+4x-x^2)+4 ?
The integral of sqrt(5+4x-x^2)+4 is 9/2 (arcsin(1/3 (x-2))+1/2 sin(2arcsin(1/3 (x-2))))+4x+C