What is the integral of 1/(5-3x) ?

The integral of 1/(5-3x) is -1/3 ln|5-3x|+C

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integral of 1/(5-3x)

\int \frac{1}{5 - 3 x} d x

- \frac{1}{3} ln ∣ 5 - 3 x ∣ + C

Solution steps

\int \frac{1}{5 - 3 x} d x

Apply u-substitution

: \int - \frac{1}{3 u} d u

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{3} ln ∣ u ∣

Substitute back

u = 5 - 3 x

= - \frac{1}{3} ln ∣ 5 - 3 x ∣

Add a constant to the solution

= - \frac{1}{3} ln ∣ 5 - 3 x ∣ + C

\frac{d}{d x} (- e^{- x} + \frac{1}{3})

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\int \frac{1}{( x - 6 ) x ^{2} - 12 x} d x

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\int_{1}^{8} x^{- \frac{5}{3}} d x