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

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

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

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

ln ∣ 1 + 2 x ∣ + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{2 u} d u

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{2} ln ∣ u ∣

Substitute back

u = 1 + 2 x

= 2 \cdot \frac{1}{2} ln ∣ 1 + 2 x ∣

Simplify

2 \cdot \frac{1}{2} ln ∣ 1 + 2 x ∣ : ln ∣ 1 + 2 x ∣

= ln ∣ 1 + 2 x ∣

Add a constant to the solution

= ln ∣ 1 + 2 x ∣ + C

\frac{d}{d y} (\frac{3 x}{2 y})

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

y = - x, y = 2 - x, y = 0

y = \frac{3 x ^{2}}{2 x + 4}

\int 2 x sin^{2} (x) d x