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

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

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integral of 3/(1-x)

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

- 3 ln ∣ 1 - x ∣ + C

Solution steps

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

Take the constant out:

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

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

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 = 1 - x

= 3 (- ln ∣ 1 - x ∣)

Simplify

= - 3 ln ∣ 1 - x ∣

Add a constant to the solution

= - 3 ln ∣ 1 - x ∣ + C

\frac{d}{d x} (x^{2} - 4 x + 3)

f (x) = \frac{( e ^{x} )}{x ^{2} + 1}

x \to 3 lim (\frac{5 x}{6 - 2 x})

\int \frac{( x ^{2} )}{9 - x ^{2}} d x

x \to 0 lim (\frac{1}{x ^{3}})