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

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

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

integral

\frac{2 x}{( 1 + x )}

2 (1 + x - ln ∣ 1 + x ∣) + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{u - 1}{u} d u

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

Expand

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

= 2 \cdot \int 1 - \frac{1}{u} 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

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

\int 1 d u = u

\int \frac{1}{u} d u = ln ∣ u ∣

= 2 (u - ln ∣ u ∣)

Substitute back

u = 1 + x

= 2 (1 + x - ln ∣ 1 + x ∣)

Add a constant to the solution

= 2 (1 + x - ln ∣ 1 + x ∣) + C

\frac{8 x}{( x ^{4} + 1 )}

\frac{1}{( 1 + x ^{2} ) ^{3}}

csc^{4} (x)

\frac{x ^{3}}{( x ^{2} + 1 )}

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

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