What is the integral of 3/(1000+2t) ?

The integral of 3/(1000+2t) is 3/2 ln|1000+2t|+C

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integral of 3/(1000+2t)

\int \frac{3}{1000 + 2 t} d t

\frac{3}{2} ln ∣ 1000 + 2 t ∣ + C

Solution steps

\int \frac{3}{1000 + 2 t} d t

Take the constant out:

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

= 3 \cdot \int \frac{1}{1000 + 2 t} d t

Apply u-substitution

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

= 3 \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

= 3 \cdot \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 \cdot \frac{1}{2} ln ∣ u ∣

Substitute back

u = 1000 + 2 t

= 3 \cdot \frac{1}{2} ln ∣ 1000 + 2 t ∣

Simplify

3 \cdot \frac{1}{2} ln ∣ 1000 + 2 t ∣ : \frac{3}{2} ln ∣ 1000 + 2 t ∣

= \frac{3}{2} ln ∣ 1000 + 2 t ∣

Add a constant to the solution

= \frac{3}{2} ln ∣ 1000 + 2 t ∣ + C

x \to 0 + lim (\frac{x - 2}{x})

\frac{d}{d x} (cos (2 x) + sin (2 x) + \frac{1}{5} e^{x})

\frac{s}{( s - 3 ) ( s - 4 ) ( s - 12 )}

\frac{\partial}{\partial x} (2 x^{\frac{3}{2}})

y = - x^{2} + 6 x, y = x^{2} - 2 x

What is the integral of 3/(1000+2t) ?
The integral of 3/(1000+2t) is 3/2 ln|1000+2t|+C