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

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

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integral of ((x-1)^2)/2

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

\frac{1}{2} (\frac{x ^{3}}{3} - x^{2} + x) + C

Solution steps

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

Take the constant out:

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

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

Expand

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

= \frac{1}{2} \cdot \int x^{2} - 2 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

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

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

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

\int 1 d x = x

= \frac{1}{2} (\frac{x ^{3}}{3} - x^{2} + x)

Add a constant to the solution

= \frac{1}{2} (\frac{x ^{3}}{3} - x^{2} + x) + C

1 + t

\frac{\partial}{\partial x} (3 x^{3} + y^{3} - 12 x y)

x \to - 2 lim (7 x + 4)

x \to 8 - lim (\frac{5}{( x - 8 ) ^{2}})

\frac{q x + r}{s x + t}

What is the integral of ((x-1)^2)/2 ?
The integral of ((x-1)^2)/2 is 1/2 ((x^3)/3-x^2+x)+C