Mathematics

Problem

A student attempts to evaluate the integral

\int x(1+x^{2})\,e^{\,(1+x^{2})^{2}}\,dx

by first letting

u=1+x^{2}

. After this substitution the new integrand still involves

e^{u^{2}}

. Which of the following best explains why making a second substitution is advantageous at this point? a) Because one can integrate

e^{u^{2}}

directly using

\displaystyle \int e^{g(u)}\,du = \tfrac{e^{g(u)}}{g'(u)}

, so introducing a second variable is unnecessary. b) Because after the first substitution the integrand becomes

\tfrac12 u e^{u^{2}}

and already matches the pattern

f'(u)e^{f(u)}

, implying the antiderivative is simply

e^{u^{2}}+C

without another substitution. c) Because

e^{u^{2}}

is a composite function whose inner expression

u^{2}

prevents direct integration; letting

v=u^{2}

changes the integral to the standard form

\displaystyle \int e^{v}\,dv

.},{ d) Because replacing

u

with

\ln v

in a second substitution would turn the exponential into a logarithm, which is easier to integrate than an exponential.