Calculate the integral.

${\int}_{-a}^{1}({x}^{2}+x)dx$

ddaeeric
2021-10-31
Answered

Calculate the integral.

${\int}_{-a}^{1}({x}^{2}+x)dx$

You can still ask an expert for help

SkladanH

Answered 2021-11-01
Author has **80** answers

Given

The given integral${\int}_{-a}^{1}({x}^{2}+x)dx$

Property used:

$\int (f\left(x\right)\pm g\left(x\right))dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx$

Calculation

${\int}_{-a}^{1}({x}^{2}+x)dx={\int}_{-a}^{1}\left({x}^{2}\right)dx+{\int}_{-a}^{1}\left(x\right)dx$

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

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

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

On further simplification,

$\int}_{-a}^{1}({x}^{2}+x)dx=\frac{1}{3}+\frac{{a}^{3}}{3}+\frac{1}{2}-\frac{{a}^{2}}{2$

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

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

The given integral

Property used:

Calculation

On further simplification,

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