What is the integral of x/(3x^2) ?

The integral of x/(3x^2) is 1/3 ln|x|+C

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integral of x/(3x^2)

\int \frac{x}{3 x ^{2}} d x

\frac{1}{3} ln ∣ x ∣ + C

Solution steps

\int \frac{x}{3 x ^{2}} d x

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{x}{x ^{2}} d x

Cancel the common factor:

x

= \frac{1}{3} \cdot \int \frac{1}{x} d x

Use the common integral:

\int \frac{1}{x} d x = ln (∣ x ∣)

= \frac{1}{3} ln ∣ x ∣

Add a constant to the solution

= \frac{1}{3} ln ∣ x ∣ + C

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