Question

Evaluate ${\int }_{-1/2}^1(x^7-x^3+x-1)dx$

• A. ${\int }_{1/2}^1{x}^7-x^3+x-1dx$
• B. ${\int }_1^{1/2}{x}^7-x^3+x-1dx$
• C. $-{\int }_{1/2}^1{x}^7-x^3+x-1dx$
• D. None of these

Answer 1

The correct option is A ${\int }_{1/2}^1{x}^7-x^3+x-1dx$

Given : ${\int }_{-\frac{1}{2}}^{\frac{1}{2}}(x^7-x^3+x-1)dx$

As we know that $x^7-x^3+x$ is an odd function

${\int }_{-\frac{1}{2}}^{\frac{1}{2}}(x^7-x^3+x)=0$

Then,

${\int }_{-\frac{1}{2}}^{\frac{1}{2}}(x^7-x^3+x-1)dx={\int }_{\frac{1}{2}}^1(x^7-x^3+x)dx-{\int }_{-\frac{1}{2}}^{1}1.dx$