Hmmm... the answer is, yes, but we have to
use a different technique. As you can see
the method of substitution will not work on
this one. That's why we need another
technique called integration by parts.
Now, let us apply that new rule to solve the previous
First, we need to decide which one is going to be f and g'.
The question is how to decide which function will be f ?
The answer is, we just need to remember this mnemonic.
Now, according to that mnemonic we will have this:
Then, we just need to find out the missing pieces required to
use the rule of integration by parts. We still need to find f '
and g, so by finding the derivative of f and antidifferentiating
g we get this:
Now, we are set. We have all the needed pieces to make
the rule work. All we need is to substitute the functions.
Now all we need is to antidifferentiate this part
then we will get our answer, which is:
After that we practiced on more
questions that use application of
the rule of integration by parts. In
some cases we need to use it more
than once, but I'm not going to
bother explaining those because
there is nothing new, except that
you just need to do use the rule
The rule can be used in
antidifferentiating a composite
function but, it can also be applied
when integrating a single function
by considering the function g'(x) to
be the factor 1.
The only thing different here is that we just used
function g'(x) to be the factor 1, in order to have two