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

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

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

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

ln ∣ 2 x + 1 ∣ + \frac{1}{2 x + 1} + C

Solution steps

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

Apply u-substitution

: \int \frac{u - 1}{4 u ^{2}} d u

= 4 \cdot \int \frac{u - 1}{4 u ^{2}} d u

Take the constant out:

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

= 4 \cdot \frac{1}{4} \cdot \int \frac{u - 1}{u ^{2}} d u

Expand

\frac{u - 1}{u ^{2}} : \frac{1}{u} - \frac{1}{u ^{2}}

= 4 \cdot \frac{1}{4} \cdot \int \frac{1}{u} - \frac{1}{u ^{2}} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 4 \cdot \frac{1}{4} (\int \frac{1}{u} d u - \int \frac{1}{u ^{2}} d u)

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{1}{u ^{2}} d u = - \frac{1}{u}

= 4 \cdot \frac{1}{4} (ln ∣ u ∣ - (- \frac{1}{u}))

Substitute back

u = 2 x + 1

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

Simplify

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

= ln ∣ 2 x + 1 ∣ + \frac{1}{2 x + 1}

Add a constant to the solution

= ln ∣ 2 x + 1 ∣ + \frac{1}{2 x + 1} + C

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