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Popular Calculus >

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

Solution

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

Solution

\frac{1}{3}

+1

Decimal

0.33333 \dots

Solution steps

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

Expand

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

= \int_{0}^{1} 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

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

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

\int_{0}^{1} 2 x d x = 1

\int_{0}^{1} 1 d x = 1

= \frac{1}{3} - 1 + 1

Simplify

= \frac{1}{3}

Graph

Popular Examples

integral from 0 to 1 of-x^2+x

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

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

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

integral from 1 to 5 of sqrt(3x+1)

\int_{1}^{5} 3 x + 1 d x

integral from 0 to 15/16 of sqrt(1+16x)

\int_{0}^{\frac{15}{16}} 1 + 16 x d x

integral from 1 to infinity of x^1

\int_{1}^{\infty} x^{1} d x

Frequently Asked Questions (FAQ)

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