Get the answer to "Calculate the integral. ∫−a1(x2+x)dx"

ddaeeric

Answered question

2021-10-31

Calculate the integral.

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

SkladanH

Skilled2021-11-01Added 80 answers

Given
The given integral

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

Property used:

\int (f (x) \pm g (x)) d x = \int f (x) d x \pm \int g (x) d x

Calculation

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

= {[\frac{x^{3}}{3}]}_{- a}^{1} + {[\frac{x^{2}}{2}]}_{- a}^{1}

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

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

On further simplification,

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

= \frac{2 + 2 a^{3} + 3 - 3 a^{2}}{6}

= \frac{2 a^{3} - 3 a^{2} + 5}{6}

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