Question

Which of the following could be the possible value(s) of ${\int }_0^1\sqrt{\left(1+x\right)\left(1+x^3\right)}\mathrm{dx}$

• A. 0.5
• B. 1
• C. 1.5
• D. 2
• E. 2.5

Answer 1

Answer: (A), (B)

1

${\int }_0^1\sqrt{\left(1+x\right)\left(1+x^3\right)}\mathrm{dx}\le \sqrt{\left({\int }_0^1\left(1+x\right)\mathrm{dx}\right)\left({\int }_0^1\left(1+x^3\right)\mathrm{dx}\right)}=\sqrt{{\left(x+\frac{x^2}{2}\right)}_0^1{\left(x+\frac{x^4}{4}\right)}_0^1}=\sqrt{\left(\frac{3}{2}\right)\left(\frac{5}{4}\right)}=\sqrt{\frac{15}{8}}\therefore {\int }_0^1\sqrt{\left(1+x\right)\left(1+x^3\right)}\mathrm{dx}\le \sqrt{\frac{15}{8}}<\sqrt{2}=1.41\mathrm{Hence},\mathrm{possible}\mathrm{values}\mathrm{can}\mathrm{be}0.5,1$