What is the integral of (2-8x)/(sqrt(1-4x^2)) ?

The integral of (2-8x)/(sqrt(1-4x^2)) is arcsin(2x)+2sqrt(1-4x^2)+C

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integral of (2-8x)/(sqrt(1-4x^2))

\int \frac{2 - 8 x}{1 - 4 x ^{2}} d x

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

Solution steps

\int \frac{2 - 8 x}{1 - 4 x ^{2}} d x

Expand

\frac{2 - 8 x}{1 - 4 x ^{2}} : \frac{2}{1 - 4 x ^{2}} - \frac{8 x}{1 - 4 x ^{2}}

= \int \frac{2}{1 - 4 x ^{2}} - \frac{8 x}{1 - 4 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 - 4 x ^{2}} d x - \int \frac{8 x}{1 - 4 x ^{2}} d x

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

\int \frac{8 x}{1 - 4 x ^{2}} d x = - 2 1 - 4 x^{2}

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

Simplify

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

Add a constant to the solution

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

x \to 1 lim (3 x^{3} + 2 x^{2} + 4)

\int \frac{x ^{5}}{1 - x ^{2}} d x

4^{x + 1}

x \to 1 lim (2 x^{3} - 4 x^{2} + x + 7)

\frac{\partial}{\partial x} (cos (2 y))

What is the integral of (2-8x)/(sqrt(1-4x^2)) ?
The integral of (2-8x)/(sqrt(1-4x^2)) is arcsin(2x)+2sqrt(1-4x^2)+C