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

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

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

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

\frac{1}{2} (ln ∣ x ∣ + x) + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{1 + x}{x} d x

Expand

\frac{1 + x}{x} : \frac{1}{x} + 1

= \frac{1}{2} \cdot \int \frac{1}{x} + 1 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

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

\int \frac{1}{x} d x = ln ∣ x ∣

\int 1 d x = x

= \frac{1}{2} (ln ∣ x ∣ + x)

Add a constant to the solution

= \frac{1}{2} (ln ∣ x ∣ + x) + C

\frac{7}{x ^{3}}

10 sin (3 x)

\frac{d y}{d x}, \frac{x}{y} = 9

\frac{d y}{d x} = y sin (x)

\int \frac{e ^{2 x}}{2 + e ^{2 x}} d x

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