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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Add a constant to the solution

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

\int \frac{9 x}{x ^{4} + 1} d x

\int 9 - 7 x^{2} d x

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

\int x e^{5 x + 2} d x

\int \frac{1}{9 x ^{4} + x ^{2}} d x

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