I am trying to find the definite integral of tanx^3, with pi/6 at the top of the integral sign and -pi/6 on the bottom.
The only problem I have is that I do not see how this fits the profile of:
f(g(x))*g'(x)
What would the g(x) be in tanx^3? Once I have that I have no problems finding the rest.
No...
But we need to distribute it through...
thus, we get
Integrating we have
Focusing on the first integral...
Let [note that the value for u is also our g(x)]. Thus, [again, this is the same as g'(x)].
Thus, what we are left with then is
Now the second integral...
Rewriting the integrand you have
Let [this is our g(x)]. Thus, [this is g'(x)].
However, this doesn't quite match the g'(x) value in the integrand . To fix this, multiply by
Thus, we see the integral become .
Combining the two, we see
Now evaluate from to
Does this make a little more sense?