What is the integral of (3x+1)/(x(x+1)) ?

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

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integral of (3x+1)/(x(x+1))

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

2 ln ∣ x + 1 ∣ + ln ∣ x ∣ + C

Solution steps

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

Expand

\frac{3 x + 1}{x ( x + 1 )} : \frac{3 x}{x ( x + 1 )} + \frac{1}{x ( x + 1 )}

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

\int \frac{3 x}{x ( x + 1 )} d x = 3 ln ∣ x + 1 ∣

\int \frac{1}{x ( x + 1 )} d x = ln ∣ x ∣ - ln ∣ x + 1 ∣

= 3 ln ∣ x + 1 ∣ + ln ∣ x ∣ - ln ∣ x + 1 ∣

Simplify

= 2 ln ∣ x + 1 ∣ + ln ∣ x ∣

Add a constant to the solution

= 2 ln ∣ x + 1 ∣ + ln ∣ x ∣ + C

x \to 0 + lim (cot (8))

x^{2} cos^{4} (x)

\int_{1}^{3} (4 z - z^{2}) d z

y^{^{'}} + 4 y = 2 x

\int \frac{x ^{2}}{( 169 + x ^{2} ) ^{2}} d x

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