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

The integral of sqrt(x^2-2x+1) is (x^2)/2-x+C

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integral of sqrt(x^2-2x+1)

\int x^{2} - 2 x + 1 d x

\frac{x ^{2}}{2} - x + C

Solution steps

\int x^{2} - 2 x + 1 d x

Factor

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

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

(x - 1)^{2} = ((x - 1)),

assuming

(x - 1) \geq 0

= \int x - 1 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

= \int x d x - \int 1 d x

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

\int 1 d x = x

= \frac{x ^{2}}{2} - x

Add a constant to the solution

= \frac{x ^{2}}{2} - x + C

\frac{d}{d x} (x \cdot e^{x^{2}})

\int \frac{cot ( x )}{sin ( x )} d x

csc (e^{2 x})

\frac{1}{p - p ^{2}} d p = d t

5 x - x^{2}, x, [0, 4]

What is the integral of sqrt(x^2-2x+1) ?
The integral of sqrt(x^2-2x+1) is (x^2)/2-x+C