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

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

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integral of (2x)/((x^2-a))

integral

\frac{2 x}{( x ^{2} - a )}

ln x^{2} - a + C

Solution steps

\int \frac{2 x}{x ^{2} - a} d x

Take the constant out:

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

= 2 \cdot \int \frac{x}{x ^{2} - a} d x

Apply u-substitution

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

= 2 \cdot \int \frac{1}{2 u} d u

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{2} ln ∣ u ∣

Substitute back

u = x^{2} - a

= 2 \cdot \frac{1}{2} ln x^{2} - a

Simplify

2 \cdot \frac{1}{2} ln x^{2} - a : ln x^{2} - a

= ln x^{2} - a

Add a constant to the solution

= ln x^{2} - a + C

(x + 9)^{4}

x sin (\frac{1}{x})

\frac{1 - 1}{x ^{4}}^{5}

\frac{( e ^{2} \cdot y )}{( 4 - y )}

x^{3} (e^{- x})^{4}

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