Answer
a) False b) False c) True
Work Step by Step
Differentiate each option to check the original integral:
a) We have: $\dfrac{d}{dx}[\sqrt {x^2+x +C}]=\dfrac{1}{2\sqrt {x^2+x +C}} (2x+1) \ne \sqrt {(2x+1)}$
b) We have: $\dfrac{d}{dx}[\sqrt {x^2+x }+C]=\dfrac{1}{2\sqrt {x^2+x}} (2x+1) \ne \sqrt {(2x+1)}$
c) We have: $\dfrac{d}{dx}[\dfrac{1}{3} (\sqrt {(2x+1)^3}+C]=\dfrac{1}{3}(3) (\sqrt {(2x+1)^2})\dfrac{1}{2 \sqrt{2x+1}} \ne \sqrt {(2x+1)}$
Hence, a) False b) False c) True
