Posted by **Joshua** on Saturday, May 2, 2009 at 4:14pm.

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?

- calculus -
**Reiny**, Saturday, May 2, 2009 at 8:06pm
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

- calculus -
**Joshua**, Sunday, May 3, 2009 at 1:48pm
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.

- calculus -
**Joshua**, Sunday, May 3, 2009 at 3:14pm
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!

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