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

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

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

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

ln ∣ x ∣ - 2 x + C

Solution steps

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

Expand

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

\int 2 d x = 2 x

= ln ∣ x ∣ - 2 x

Add a constant to the solution

= ln ∣ x ∣ - 2 x + C

y = x^{3}, y = 1, x = 2

n = 0 \sum \infty \frac{3 ^{n}}{4 ^{n}}

n = 1 \sum \infty \frac{2 n + 1}{n ^{4}}

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

x \to \infty lim (f (x) (x)^{2})