Question

Evalaute: ${\int }_{-1}^1(\sqrt{1+x+x^2}-\sqrt{1-x+x^2})dx$

• A. $0$
• B. $1$
• C. $-1$
• D. $3$

Answer 1

The correct option is A $0$

$I={\int }_{-1}^1(\sqrt{1+x+x^2}-\sqrt{1-x+x^2})dx$

Using the property ${\int }_a^{b}f\left(x\right)dx={\int }_a^{b}f(a+b-x)dx$

$I={\int }_{-1}^1(\sqrt{1+(-x)+(-x{)}^2}-\sqrt{1-(-x)+(-x{)}^2})dx$

$I={\int }_{-1}^1(\sqrt{1-x+x^2}-\sqrt{1+x+x^2})dx$

$⟹I=-I$

$⟹2I=0$

$⟹I=0$

Hence, answer is option-(A).