Evaluate the definite integral using integration by parts with Way 1.
We found that
Evaluate the definite integral using integration by parts with Way 2.
Then apply the formula, keeping the limits of integration all the way through:
For the sakes of tidiness and sanity, let's work out each of these pieces by itself.
Now we put things back together:
Let's work out the first term:
And now the second term:
Apply the formula, keeping the limits of integration around:
Work out the first chunk:
And now work out the rest:
We need to use the factor trick on this one. If we rewrite the integrand as
it's easier to see how to take the parts.
u = x2
v' = 2xex2
u' = 2x
v = ex2
Then we apply the formula:
Work out the first bit:
Now work out the new integral:
u = x
v' = ex
u' = 1
v = ex
Apply the formula:
If we want, we can play with the logs a little to make this prettier.
Make it rain.