What is the integral from x to 1 of (y-1)^2 ?

The integral from x to 1 of (y-1)^2 is x^2-x+(-x^3+1)/3

Popular >

integral from x to 1 of (y-1)^2

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

x^{2} - x + \frac{- x ^{3} + 1}{3}

Solution steps

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

Expand

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

= \int_{x}^{1} y^{2} - 2 y + 1 d y

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

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

\int_{x}^{1} 2 y d y = 1 - x^{2}

\int_{x}^{1} 1 d y = 1 - x

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

Simplify

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

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

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

\int_{1 5^{\circ}}^{3 0^{\circ}} x^{2} sin (5) x^{3} d x

\int_{- 1}^{1} \frac{x}{2} d x

\int_{- 1}^{- 2} f (t)

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