What is the integral of 1/(4-5x) ?

The integral of 1/(4-5x) is -1/5 ln|4-5x|+C

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integral of 1/(4-5x)

\int \frac{1}{4 - 5 x} d x

- \frac{1}{5} ln ∣ 4 - 5 x ∣ + C

Solution steps

\int \frac{1}{4 - 5 x} d x

Apply u-substitution

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

= \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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{5} ln ∣ u ∣

Substitute back

u = 4 - 5 x

= - \frac{1}{5} ln ∣ 4 - 5 x ∣

Add a constant to the solution

= - \frac{1}{5} ln ∣ 4 - 5 x ∣ + C

\frac{\partial}{\partial x} (x^{3})

\frac{\partial}{\partial x} ((3 x^{2} + 5 x + 1) \cdot (y + 4))

\frac{d y}{d x} = y (x y^{3} - 1)

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

y^{^{''}} + 4 y^{^{'}} + 5 y = 35 e - 4 x, y (0) = - 5, y^{^{'}} (0) = 1