Let F(x) = ∫ 12x (sqrt (t^2 + t)) dt Find the domain of F in terms of x.

Question

Let F(x) = ∫₁^2x (sqrt (t^2 + t)) dt

Find the domain of F in terms of x.

Answer 1

To solve this, let's look at the integrand. Over the real numbers, the domain of sqrt(t^2+t) is all values of t outside the interval (-1,0). Therefore, the limits of integration must be outside the interval (-1,0). What values of x make this statement true?

Note that if we change the order of integration, we have

F(x) = -\int_{2x}^{1} sqrt(t^2+t) dt. Therefore, we must have 2x>0, or simply x>0, which would be the domain.