What can be considered as a constant in integration? Suppose I have an integral to evaluate, which is of the form ,
$$\int_0^x xf(t)\,dt$$
So , is it correct to consider the $x$ in the integrand as a constant and remove it outside the integral? I think the answer is yes because it does not change with $t$ . But I’m still confused as the upper limit also contains $x$ . Please help me with this .
 A: If you just want to calculate the value of the integral, this should be fine if I'm not mistaken, as you might also call the upper bound $u$ and later set $u=x$ again after calculation. More precisely, to evaluate the integral, I'm parametrizing the upper bound to $u$ s.t. we consider
$$h(u)=\int_0^uxf(t)dt=x\int_0^uf(t)dt$$
Thus, you may evaluate $\int_0^uf(t)dt$ in dependence on $u$ and then derive the value of the original integral via $h(x)$.
However, if you are working with the by the integral induced function
$$g(x)=\int_0^xxf(t)dt$$
and want to perform more complex tasks like differentiation or integration of $g$, then this is called a parametric integral and the answer is no. But in this case, $x$ would also less be considered a constant value than a variable(for this respective function).
A: 
...[I]s it correct to consider the $x$ in the integrand as a constant and remove it outside the integral?

Well, the answer depends on how you regard $x,$ depending on the problem you want to solve. Since you want to evaluate the integral, it would seem that $x$ should be regarded as some (as yet unspecified, but fixed) number. In this case you then proceed as usual. It is then clear that for each $x$ (regarded as a number) for which the integral is defined, there is no difference in the evaluation.
