What is the integral of 3/(10-t) ?

The integral of 3/(10-t) is -3ln|10-t|+C

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integral of 3/(10-t)

\int \frac{3}{10 - t} d t

- 3 ln ∣ 10 - t ∣ + C

Solution steps

\int \frac{3}{10 - 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}{10 - t} d t

Apply u-substitution

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

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

Take the constant out:

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

= 3 (- \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 (- ln ∣ u ∣)

Substitute back

u = 10 - t

= 3 (- ln ∣ 10 - t ∣)

Simplify

= - 3 ln ∣ 10 - t ∣

Add a constant to the solution

= - 3 ln ∣ 10 - t ∣ + C

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