What is the integral of (x^2+1)/(sqrt(4-x^2)) ?

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

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integral of (x^2+1)/(sqrt(4-x^2))

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

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

Solution steps

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

Expand

\frac{x ^{2} + 1}{4 - x ^{2}} : \frac{x ^{2}}{4 - x ^{2}} + \frac{1}{4 - x ^{2}}

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

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

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

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

Simplify

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

Add a constant to the solution

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

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