What is the integral of (15)/(400+5t) ?

The integral of (15)/(400+5t) is 3ln|400+5t|+C

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integral of (15)/(400+5t)

\int \frac{15}{400 + 5 t} d t

3 ln ∣ 400 + 5 t ∣ + C

Solution steps

\int \frac{15}{400 + 5 t} d t

Take the constant out:

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

= 15 \cdot \int \frac{1}{400 + 5 t} d t

Apply u-substitution

= 15 \cdot \int \frac{1}{5 u} d u

Take the constant out:

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

= 15 \cdot \frac{1}{5} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 15 \cdot \frac{1}{5} ln ∣ u ∣

Substitute back

u = 400 + 5 t

= 15 \cdot \frac{1}{5} ln ∣ 400 + 5 t ∣

Simplify

15 \cdot \frac{1}{5} ln ∣ 400 + 5 t ∣ : 3 ln ∣ 400 + 5 t ∣

= 3 ln ∣ 400 + 5 t ∣

Add a constant to the solution

= 3 ln ∣ 400 + 5 t ∣ + C

What is the integral of (15)/(400+5t) ?
The integral of (15)/(400+5t) is 3ln|400+5t|+C