Question
Use a change of variables to evaluate the following indefinite integral.
Determine a change of variables from
to u. Choose the correct answer below.
A.
B.
C.
D.
Write the integral in terms of
.
du
Evaluate the integral.
Determine a change of variables from
A.
B.
C.
D.
Write the integral in terms of
Evaluate the integral.
Ask by Schofield Schneider. in the United States
Mar 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
Let
. Then, the integral becomes
, which evaluates to
. Substituting back, the answer is
.
Solution
Let
. Notice that
so differentiating
with respect to
we have
which implies
Substituting into the original integral
we replace
with
and
:
This shows that option D,
, is the correct substitution.
Now, evaluating the integral:
Substituting back
, we obtain
Thus, the evaluated integral is
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Extra Insights
To evaluate the given integral using a change of variables, let’s choose
(Option D). This means that
and
. The differential
can be expressed as
.
Now, substitute everything into the integral. The integrand becomes:
Thus, we have:
Now, let’s evaluate the integral:
Finally, substituting back for
: