What is the integral of (4(x+1))/(2x+1) ?

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

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integral of (4(x+1))/(2x+1)

\int \frac{4 ( x + 1 )}{2 x + 1} d x

2 x + 1 + ln ∣ 2 x + 1 ∣ + C

Solution steps

\int \frac{4 ( x + 1 )}{2 x + 1} d x

Take the constant out:

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

= 4 \cdot \int \frac{x + 1}{2 x + 1} d x

Expand

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

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

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

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

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

Simplify

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

= 2 x + 1 + ln ∣ 2 x + 1 ∣

Add a constant to the solution

= 2 x + 1 + ln ∣ 2 x + 1 ∣ + C

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