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

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

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

integral

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

- \frac{1}{2} ln a^{2} - x^{2} + C

Solution steps

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

Apply u-substitution

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

= \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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{2} ln ∣ u ∣

Substitute back

u = a^{2} - x^{2}

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

Add a constant to the solution

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

3 e^{2} x

\frac{1}{( a ^{2} + b ^{2} )}

\frac{- 1}{t}

\frac{2 x}{e ^{x}}

x^{3} \cdot (e^{- x})^{2}

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