What is the integral from 0 to 1 of ((x+3)-(x^5+3)) ?

The integral from 0 to 1 of ((x+3)-(x^5+3)) is 1/3

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integral from 0 to 1 of ((x+3)-(x^5+3))

\int_{0}^{1} ((x + 3) - (x^{5} + 3)) d x

\frac{1}{3}

Decimal

0.33333 \dots

Solution steps

\int_{0}^{1} x + 3 - (x^{5} + 3) d x

Simplify

= \int_{0}^{1} - x^{5} + x 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^{5} d x + \int_{0}^{1} x d x

\int_{0}^{1} x^{5} d x = \frac{1}{6}

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

= - \frac{1}{6} + \frac{1}{2}

Simplify

- \frac{1}{6} + \frac{1}{2} : \frac{1}{3}

= \frac{1}{3}

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What is the integral from 0 to 1 of ((x+3)-(x^5+3)) ?
The integral from 0 to 1 of ((x+3)-(x^5+3)) is 1/3