What is the integral of (x^2-a^2)/(4x) ?

The integral of (x^2-a^2)/(4x) is 1/4 ((x^2)/2-a^2ln|x|)+C

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integral of (x^2-a^2)/(4x)

\int \frac{x ^{2} - a ^{2}}{4 x} d x

\frac{1}{4} (\frac{x ^{2}}{2} - a^{2} ln ∣ x ∣) + C

Solution steps

\int \frac{x ^{2} - a ^{2}}{4 x} d x

Take the constant out:

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

= \frac{1}{4} \cdot \int \frac{x ^{2} - a ^{2}}{x} d x

Expand

\frac{x ^{2} - a ^{2}}{x} : x - \frac{a ^{2}}{x}

= \frac{1}{4} \cdot \int x - \frac{a ^{2}}{x} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{4} (\int x d x - \int \frac{a ^{2}}{x} d x)

\int x d x = \frac{x ^{2}}{2}

\int \frac{a ^{2}}{x} d x = a^{2} ln ∣ x ∣

= \frac{1}{4} (\frac{x ^{2}}{2} - a^{2} ln ∣ x ∣)

Add a constant to the solution

= \frac{1}{4} (\frac{x ^{2}}{2} - a^{2} ln ∣ x ∣) + C

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What is the integral of (x^2-a^2)/(4x) ?
The integral of (x^2-a^2)/(4x) is 1/4 ((x^2)/2-a^2ln|x|)+C