What is the integral of 1/(x(x-2)) ?

The integral of 1/(x(x-2)) is -1/2 ln|x|+1/2 ln|x-2|+C

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integral of 1/(x(x-2))

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

- \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x - 2 ∣ + C

Solution steps

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

Complete the square

x (x - 2) : (x - 1)^{2} - 1

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

Apply u-substitution

: \int \frac{1}{u ^{2} - 1} d u

= \int \frac{1}{u ^{2} - 1} d u

Factor

u^{2} - 1 : - (- u^{2} + 1)

= \int \frac{1}{- ( - u ^{2} + 1 )} d u

Take the constant out:

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

= - \int \frac{1}{- u ^{2} + 1} d u

Use the common integral:

\int \frac{1}{- u ^{2} + 1} d u = \frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}

= - (\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2})

Substitute back

u = x - 1

= - (\frac{ln ∣ x - 1 + 1 ∣}{2} - \frac{ln ∣ x - 1 - 1 ∣}{2})

Simplify

- (\frac{ln ∣ x - 1 + 1 ∣}{2} - \frac{ln ∣ x - 1 - 1 ∣}{2}) : - \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x - 2 ∣

= - \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x - 2 ∣

Add a constant to the solution

= - \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x - 2 ∣ + C

\int \frac{6400 q}{16 q ^{2} + 3200} d q

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

x \to 5 lim (3 x^{2} - 4 x - 1)

f (x) = 8 x + x^{2} + 7 x + 3

\int_{1}^{2} y^{3} + \frac{1}{4 y ^{3}} d y

What is the integral of 1/(x(x-2)) ?
The integral of 1/(x(x-2)) is -1/2 ln|x|+1/2 ln|x-2|+C