We're doing indefinite integrals using the substitution rule right now in class.
The problem:
(integral of) (e^6x)csc(e^6x)cot(e^6x)dx
I am calling 'u' my substitution variable. I feel like I've tried every possible substitution, but I still haven't found the right one.
The most promising substitution:
u= csc(e^6x)
du/dx=-(e^6x)csc(e^6x)cot(e^6x)
du/((-e^6x)csc(e^6x)cot(e^6x))=dx
so my equation would become
(integral of) usin(e^6x)
Now I don't know what to do, because we haven't learned how to the problem like this. I feel like there must be some substitution that will leave me with only one term to integrate, but I don't think I've found it. Suggestions?
you are so close
now let's do some reverse "thinking"
it looks like you know that if
y = cscx, the dy/dx = -cscx cotx
now what about
y = csc(e^6x) ?
wouldn't dy/dx = 6e^(6x)(-csc(e^6x))(cot(e^6x))
= -6e^(6x)(csc(e^6x))(cot(e^6x)) ?
Now compare that with what was given.
the only "extra" is see is the -6 in front, and that is merely a constant, so let's fudge it.
then (integral of) (e^6x)csc(e^6x)cot(e^6x)dx
= -(1/6)csc(e^6x) + C
I know that your answer is right because it is one of the options on my homework sheet, but I don't think I quite understand how everything cancels out.
If we call csc(e^6x) y from the beginning, upon initial substitution we have:
(integral of) (e^6x)(y)(cot(e^6x))dx
now dy/dx= -6e^(6x)csc(e^6x)cot(e^6x)
However, in our integral equation we have dx, not dy/dx, so we need to rearrange this so that we can directly substitute for dx
which would give us dy/(--6e^(6x)csc(e^6x)cot(e^6x)) = dx.
If you plug that into our integral, we have
-1/6 (integral of) y * 1/csc(e^6x) dy
or -1/6 (integral of) sin(e^6x)y dy
I know this isn't right, so I feel like I don't properly understand what to do with the dy/dx situation.
Never mind, I figured it out; because y= csc(e^6x), the ys cancel out, so we are left with -1/6 (integral of) dy, which gives us -1/6 csc(e^6x) + C. Thank you for the help!
To solve the integral ∫ (e^6x)csc(e^6x)cot(e^6x)dx, you correctly started by trying to find a suitable substitution. Let's reconsider your most promising substitution:
u = csc(e^6x)
du/dx = -(e^6x)csc(e^6x)cot(e^6x)
du / (-(e^6x)csc(e^6x)cot(e^6x)) = dx
Based on this substitution, the integral can be rewritten as:
∫ usin(e^6x) dx
Now, let's continue solving the integral using a different approach. Instead of trying to find a new substitution, we will use integration by parts. The formula for integration by parts is:
∫ u dv = uv - ∫ v du
In our case, we can set:
u = u = u
dv = sin(e^6x) dx
To find du and v, we need to differentiate u and integrate dv, respectively.
du = [d(csc(e^6x))] = -csc(e^6x)cot(e^6x)dx
To find v, integrate dv:
v = ∫ sin(e^6x) dx
To integrate v, we can make a substitution of t = e^6x, then dt = e^6x dx.
v = ∫ sin(t) * (1/e^6) dt
= (1/e^6) ∫ sin(t) dt
= -(1/e^6) cos(t)
Now, we can apply the integration by parts formula:
∫ usin(e^6x) dx = (u * v) - ∫ v * du
= (u * (-1/e^6) * cos(t)) - ∫ - cos(t) * (-csc(e^6x)cot(e^6x)dx)
= -(u * cos(t) / e^6) - ∫ cos(t) * csc(e^6x)cot(e^6x) dx
Substituting back our original variable:
= -(csc(e^6x) * cos(e^6x) / e^6) - ∫ cos(e^6x) * csc(e^6x) * cot(e^6x) dx
Notice that the last term in the equation is identical to our original integral. Therefore, we can substitute the original integral into this equation:
= -(csc(e^6x) * cos(e^6x) / e^6) - ∫ (e^6x)csc(e^6x)cot(e^6x) dx
Let's represent the original integral as I:
I = ∫ (e^6x)csc(e^6x)cot(e^6x) dx
Substituting it in the equation, we obtain:
I = -(csc(e^6x) * cos(e^6x) / e^6) - I
Rearranging the equation, we get:
2I = -(csc(e^6x) * cos(e^6x) / e^6)
Finally, isolating I gives:
I = - (csc(e^6x) * cos(e^6x)) / (2 * e^6)
Therefore, the indefinite integral of (e^6x)csc(e^6x)cot(e^6x)dx is - (csc(e^6x) * cos(e^6x)) / (2 * e^6) + C, where C represents the constant of integration.