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

The integral of 1/(1/x+2) is (x^2)/(1+2x)-1/4 (ln|1+2x|+1/(1+2x))+C

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integral of 1/(1/x+2)

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

\frac{x ^{2}}{1 + 2 x} - \frac{1}{4} (ln ∣ 1 + 2 x ∣ + \frac{1}{1 + 2 x}) + C

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{x}{( 1 + 2 x ) ^{2}} d x = \frac{1}{4} (ln ∣ 1 + 2 x ∣ + \frac{1}{1 + 2 x})

= \frac{x ^{2}}{1 + 2 x} - \frac{1}{4} (ln ∣ 1 + 2 x ∣ + \frac{1}{1 + 2 x})

Add a constant to the solution

= \frac{x ^{2}}{1 + 2 x} - \frac{1}{4} (ln ∣ 1 + 2 x ∣ + \frac{1}{1 + 2 x}) + C

x \to 0 lim (5 x - 6)

2 y = 4 x, y = 4, 2 y + 4 x = 8

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

sin (3 x) + cos (8 x)

\frac{1}{3} x^{6} (x^{2} - 3)

What is the integral of 1/(1/x+2) ?
The integral of 1/(1/x+2) is (x^2)/(1+2x)-1/4 (ln|1+2x|+1/(1+2x))+C