What is the integral of (3x-1)/(sqrt(1-2x)) ?

The integral of (3x-1)/(sqrt(1-2x)) is -xsqrt(-2x+1)+C

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integral of (3x-1)/(sqrt(1-2x))

\int \frac{3 x - 1}{1 - 2 x} d x

- x - 2 x + 1 + C

Solution steps

\int \frac{3 x - 1}{1 - 2 x} d x

Expand

\frac{3 x - 1}{1 - 2 x} : \frac{3 x}{1 - 2 x} - \frac{1}{1 - 2 x}

= \int \frac{3 x}{1 - 2 x} - \frac{1}{1 - 2 x} 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{3 x}{1 - 2 x} d x - \int \frac{1}{1 - 2 x} d x

\int \frac{3 x}{1 - 2 x} d x = - (x + 1) - 2 x + 1

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

= - (x + 1) - 2 x + 1 - (- 1 - 2 x)

Simplify

- (x + 1) - 2 x + 1 - (- 1 - 2 x) : - x - 2 x + 1

= - x - 2 x + 1

Add a constant to the solution

= - x - 2 x + 1 + C

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What is the integral of (3x-1)/(sqrt(1-2x)) ?
The integral of (3x-1)/(sqrt(1-2x)) is -xsqrt(-2x+1)+C