Which of the following best explains why, when evaluating a definite integral by substituting u=g(x), the limits of integration must be rewritten in terms of u before the result can be confirmed?
(a) Because the differential du already compensates for the substitution, the original x-limits must remain in order to preserve the interval length.
(b) Without rewriting the limits, differentiating the final numerical result cannot reproduce the original integrand–value relationship, so the computation may be invalid.
(c) Because the limits need to be changed only when g′(x) becomes negative on the interval; if g′(x)>0 everywhere, the original bounds are still appropriate.
(d) Because the dummy variable of integration has no effect on the limits—the antiderivative evaluated at the original x-bounds will always give the same number.