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

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

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integral of 1/(1-3x)

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

- \frac{1}{3} ln ∣ 1 - 3 x ∣ + C

Solution steps

\int \frac{1}{1 - 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 = 1 - 3 x

= - \frac{1}{3} ln ∣ 1 - 3 x ∣

Add a constant to the solution

= - \frac{1}{3} ln ∣ 1 - 3 x ∣ + C

\int x 3 2 x d x

f (t) = \frac{5 t ^{2} + t + 6}{t}

\int \frac{7 x}{( 1 - x ^{2} ) ^{3}} d x

x \to π lim (\frac{cos ( x )}{x})

\frac{d}{d x} (\frac{2 x - 3}{x ^{2} + 1})