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

The integral of 1/(1-2x)-cos(x/3) is -1/2 ln|1-2x|-3sin(x/3)+C

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

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

- \frac{1}{2} ln ∣ 1 - 2 x ∣ - 3 sin (\frac{x}{3}) + C

Solution steps

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

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

\int cos (\frac{x}{3}) d x = 3 sin (\frac{x}{3})

= - \frac{1}{2} ln ∣ 1 - 2 x ∣ - 3 sin (\frac{x}{3})

Add a constant to the solution

= - \frac{1}{2} ln ∣ 1 - 2 x ∣ - 3 sin (\frac{x}{3}) + C

What is the integral of 1/(1-2x)-cos(x/3) ?
The integral of 1/(1-2x)-cos(x/3) is -1/2 ln|1-2x|-3sin(x/3)+C