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Answer:
Step-by-step explanation:
.
It is apparently obvious we could expand the bracket and integrate term-by-term. This method would work but is very time consuming (and you could easily make a mistake) so we use a different method: integration by substitution.
Integration by substitution involves swapping the variable for another variable which depends on x:
. (We are going to choose
for this question).
The very first step is to choose a suitable substitution. That is, an equation which is going to make the integration easier. There is a trick for spotting this however: if an integral contains both a term and it's derivative then use the substitution
.
Your integral contains the term . The derivative is
and (ignoring the constants) we see
is also in the integral and so the substitution
will unravel this integral!
Step 2: We must now swap the variable of integration from x to u. That means interchanging all the x's in the integrand (the term being integrated) for u's and also swapping (dx" to "du").
Then,
.
The substitution has made this integral is easy to solve!
Finally we can substitute back to get the answer in terms of x:
The Jacobian has determinant