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

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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

\frac{d}{d x} f (x) = x^{2} + 8 x + 7

x \to 0 lim (\frac{7 x ^{5} + 5 x ^{2} + 12 x}{3 x ^{5} + 4 x})

x^{2} - 3 x y = 10

y^{^{'}} = \frac{sec ( y )}{( x - 3 ) ^{2}}

n = 0 \sum \infty \frac{2}{1}

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