What is the integral of 5/(1000+t) ?

The integral of 5/(1000+t) is 5ln|1000+t|+C

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integral of 5/(1000+t)

\int \frac{5}{1000 + t} d t

5 ln ∣ 1000 + t ∣ + C

Solution steps

\int \frac{5}{1000 + t} d t

Take the constant out:

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

= 5 \cdot \int \frac{1}{1000 + t} d t

Apply u-substitution

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

= 5 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 5 ln ∣ u ∣

Substitute back

u = 1000 + t

= 5 ln ∣ 1000 + t ∣

Add a constant to the solution

= 5 ln ∣ 1000 + t ∣ + C

\int x^{2} (4 - x) d x

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

\frac{d}{d x} (20 x - x^{2})

\frac{d x}{d t} = \frac{3}{2} t cos^{2} (x), x (0) = - 1.4

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

What is the integral of 5/(1000+t) ?
The integral of 5/(1000+t) is 5ln|1000+t|+C