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

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

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

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

4 1 - x^{2} + arcsin (x) + C

Solution steps

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

Expand

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

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

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

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

= - (- 4 1 - x^{2}) + arcsin (x)

Simplify

= 4 1 - x^{2} + arcsin (x)

Add a constant to the solution

= 4 1 - x^{2} + arcsin (x) + C

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