What is the integral from 0 to 2 of sqrt(x^2-2x+1) ?

The integral from 0 to 2 of sqrt(x^2-2x+1) is 1

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integral from 0 to 2 of sqrt(x^2-2x+1)

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

1

Solution steps

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

Factor

x^{2} - 2 x + 1 : (x - 1)^{2}

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

Apply exponent rule:

a^{2} = ∣ a ∣

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

Eliminate Absolutes

= \int_{0}^{1} (x - 1)^{2} d x + \int_{1}^{2} (x - 1)^{2} d x

\int_{0}^{1} (x - 1)^{2} d x = \frac{1}{2}

\int_{1}^{2} (x - 1)^{2} d x = \frac{1}{2}

= \frac{1}{2} + \frac{1}{2}

Simplify

\frac{1}{2} + \frac{1}{2} : 1

= 1

What is the integral from 0 to 2 of sqrt(x^2-2x+1) ?
The integral from 0 to 2 of sqrt(x^2-2x+1) is 1