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

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

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

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

\frac{1}{2} ln ∣ 1 + 2 x ∣ + C

Solution steps

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

Apply u-substitution

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

= \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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2} ln ∣ u ∣

Substitute back

u = 1 + 2 x

= \frac{1}{2} ln ∣ 1 + 2 x ∣

Add a constant to the solution

= \frac{1}{2} ln ∣ 1 + 2 x ∣ + C

y^{^{''}} - 3 y^{^{'}} - 10 y = - 4 e^{4 t}, y (0) = 0, y^{^{'}} (0) = 0

\int 12 e^{4 x} d x

10 t^{3} e^{- 2 t}

y = \frac{1}{2} x^{3} + 2, y = x + 1, 0, 2

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