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

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

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integral of 2/(1+x)

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

2 ln ∣ 1 + x ∣ + C

Solution steps

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

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 ln ∣ u ∣

Substitute back

u = 1 + x

= 2 ln ∣ 1 + x ∣

Add a constant to the solution

= 2 ln ∣ 1 + x ∣ + C

y = x^{3}

\int_{- 1}^{1} (y^{2} - 2) - (e^{y}) d y

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

x \to \infty lim (t^{x})

x \to 4 lim (x^{2} - 4 x)