What is the integral from 0 to 1 of (|3x-1|-2|x|) ?

The integral from 0 to 1 of (|3x-1|-2|x|) is -1/6

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integral from 0 to 1 of (|3x-1|-2|x|)

\int_{0}^{1} (∣ 3 x - 1 ∣ - 2 ∣ x ∣) d x

- \frac{1}{6}

Decimal

- 0.16666 \dots

Solution steps

\int_{0}^{1} ∣ 3 x - 1 ∣ - 2 ∣ x ∣ d x

Eliminate Absolutes

= \int_{0}^{\frac{1}{3}} - 3 x + 1 - 2 x d x + \int_{\frac{1}{3}}^{1} 3 x - 1 - 2 x d x

\int_{0}^{\frac{1}{3}} - 3 x + 1 - 2 x d x = \frac{1}{18}

\int_{\frac{1}{3}}^{1} 3 x - 1 - 2 x d x = - \frac{2}{9}

= \frac{1}{18} - \frac{2}{9}

Simplify

\frac{1}{18} - \frac{2}{9} : - \frac{1}{6}

= - \frac{1}{6}

What is the integral from 0 to 1 of (|3x-1|-2|x|) ?
The integral from 0 to 1 of (|3x-1|-2|x|) is -1/6