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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Add a constant to the solution

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

x \to + 4 + lim (2 x - 7)

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

\int \frac{1}{4 e ^{- 8 x} + e ^{8 x}} d x

\frac{\partial}{\partial x} (x (x^{2} + y^{2} + z^{2} - 1))

x \to 1 - lim (x^{2} - 5)

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