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

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

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

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

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

Solution steps

\int \frac{x}{x ^{2} + a ^{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 = x^{2} + a^{2}

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

Add a constant to the solution

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

f (x) = 9 - x^{2}

y = x^{2} ln (4 x + 3)

\frac{d}{d x} ((x^{2})^{x})

\int (2 y) d y

\int \frac{( 5 x - 7 )}{( x ^{2} - x - 2 ) ( x - 3 )} d x

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