What is the integral of |5x| ?

The integral of |5x| is 5/2 x|x|+C

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integral of |5x|

\int ∣ 5 x ∣ d x

\frac{5}{2} x ∣ x ∣ + C

Solution steps

\int ∣ 5 x ∣ d x

Apply absolute rule:

∣ a x ∣ = a ∣ x ∣, a \geq 0

= \int 5 ∣ x ∣ d x

Take the constant out:

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

= 5 \cdot \int ∣ x ∣ d x

Use the common integral:

\int ∣ x ∣ d x = \frac{x ∣ x ∣}{2}

= 5 \cdot \frac{x ∣ x ∣}{2}

Simplify

5 \cdot \frac{x ∣ x ∣}{2} : \frac{5}{2} x ∣ x ∣

= \frac{5}{2} x ∣ x ∣

Add a constant to the solution

= \frac{5}{2} x ∣ x ∣ + C

f (x) = \frac{53}{0.01 x ^{2} + 1}

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