Question

find integral of -2 to 0 of x at f(x^2) dx if f(x) dx=1

Answer 1

Using an appropriate substitution, you can see that
\[\int_{-2}^0xf(x^2)\,\mathrm{d}x =\frac{1}{2}\int_{-2}^0f(x^2)\,\mathrm{d}(x^2)\color{lightgray}{=\frac{1}{2}\int_4^0f(t)\,\mathrm{d}t=-\frac{1}{2}\int_0^4f(t)\,\mathrm{d}t}\]I'm not sure what to make of the hypothesis, however. What do you mean by "... if f(x) dx=1"? I assume it's the value of the integral of \(f\) but over what interval?