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

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

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

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

2 arcsin (x) - 1 - x^{2} + C

Solution steps

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

Expand

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

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

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

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

= 2 arcsin (x) - 1 - x^{2}

Add a constant to the solution

= 2 arcsin (x) - 1 - x^{2} + C

x \to 3 + lim (\frac{1}{( x - 3 ) ^{2}})

\int \frac{sin ( 5 x )}{1 - cos ( 5 x )} d x

\frac{d y}{( 1 + 4 y ^{2} )} = - \frac{x}{( 1 + x ^{4} )} d x

6 x^{\frac{3}{2}}

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