What is the integral of (30)/(0.06t+5) ?

The integral of (30)/(0.06t+5) is 500ln|0.06t+5|+C

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integral of (30)/(0.06t+5)

\int \frac{30}{0.06 t + 5} d t

500 ln ∣ 0.06 t + 5 ∣ + C

Solution steps

\int \frac{30}{0.06 t + 5} d t

Take the constant out:

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

= 30 \cdot \int \frac{1}{0.06 t + 5} d t

Apply u-substitution

: \int \frac{1}{0.06 u} d u

= 30 \cdot \int \frac{1}{0.06 u} d u

Take the constant out:

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

= 30 \cdot \frac{1}{0.06} \cdot \int \frac{1}{u} d u

Simplify

= 30 \cdot 16.66666 \dots \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 30 \cdot 16.66666 \dots ln ∣ u ∣

Substitute back

u = 0.06 t + 5

= 30 \cdot 16.66666 \dots ln ∣ 0.06 t + 5 ∣

Simplify

= 500 ln ∣ 0.06 t + 5 ∣

Add a constant to the solution

= 500 ln ∣ 0.06 t + 5 ∣ + C

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What is the integral of (30)/(0.06t+5) ?
The integral of (30)/(0.06t+5) is 500ln|0.06t+5|+C