What is the integral of 3/(5y-4) ?

The integral of 3/(5y-4) is 3/5 ln|5y-4|+C

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integral of 3/(5y-4)

\int \frac{3}{5 y - 4} d y

\frac{3}{5} ln ∣ 5 y - 4 ∣ + C

Solution steps

\int \frac{3}{5 y - 4} d y

Take the constant out:

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

= 3 \cdot \int \frac{1}{5 y - 4} d y

Apply u-substitution

: \int \frac{1}{5 u} d u

= 3 \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

= 3 \cdot \frac{1}{5} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 \cdot \frac{1}{5} ln ∣ u ∣

Substitute back

u = 5 y - 4

= 3 \cdot \frac{1}{5} ln ∣ 5 y - 4 ∣

Simplify

3 \cdot \frac{1}{5} ln ∣ 5 y - 4 ∣ : \frac{3}{5} ln ∣ 5 y - 4 ∣

= \frac{3}{5} ln ∣ 5 y - 4 ∣

Add a constant to the solution

= \frac{3}{5} ln ∣ 5 y - 4 ∣ + C

x \to 0 lim (\frac{6 x n - 5 x n - 1 - 7 x n - 2}{- 6 x n - 4 x n - 2})

f (x) = (x^{4} + 3 x^{2} - 2)^{5}

ln (s + 1)

\int \frac{3 - x}{x - 4} d x

(0, 2), (4, - 1)