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

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

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integral of 2/(2+2x)

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

ln ∣ 2 + 2 x ∣ + C

Solution steps

\int \frac{2}{2 + 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}{2 + 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 = 2 + 2 x

= 2 \cdot \frac{1}{2} ln ∣ 2 + 2 x ∣

Simplify

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

= ln ∣ 2 + 2 x ∣

Add a constant to the solution

= ln ∣ 2 + 2 x ∣ + C

f (x) = - 5 x^{2} + 6, at x = 5

x \to - 3 lim (4 x + 28 - 4)

\frac{d}{d x} (\frac{7 x}{2 x ^{3}})

\frac{d}{d x} ((x^{3} + 2)^{2} (x^{5} + 4)^{4})