Which of the following best explains why

MATHEMATICS · Single correct · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Which of the following best explains why, when evaluating

\displaystyle \int_{0}^{1} 2x e^{x^{2}}\,dx

, a student might choose to find an antiderivative in terms of

x

and then substitute the original limits instead of changing the limits to match the substitution

u=x^{2}

? (a) Because computing the new limit values would require solving

u=x^{2}

for

x

, introducing square roots the student prefers to avoid (b) Because the Fundamental Theorem of Calculus only applies if the limits match the variable of integration, so limits cannot be changed (c) Because the antiderivative of

e^{x^{2}}

is believed to lack an elementary form, so the student keeps the limits in

x

instead of switching to

u

(d) Because changing limits is only necessary for *indefinite* integrals; for definite integrals one must return to the original variable before using the limits

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