#$&*
Mth 174
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
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I am studying for my test #1, I have some questions that i don't know how to do.
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first one, Find the general antiderivative of sin(x) / [ 8 + cos^ 8 (x) ].
@& If you let u = cos(x) then du = - sin(x) dx. The integral becomes -du / (1 + u^8).
I would expect to see this attempt.
However at this point you're pretty much stuck. The denominator factors, but not over real numbers, so the techniques of integration are beyond those available in this course.
Had the denominator factored over the real numbers, then partial fractions would become a possibility.*@
and second, Determine whether the antiderivatives of e^( 4 x) / (1 + e^ 4 x) and cos( 4 x) / (1 + cos( 4 x)) are in fact different expressions of the same problem. If so specify the problem; if not state why not.
@& If you let u = e^(4 x) then
integral(e^( 4 x) / (1 + e^ 4 x) dx
becomes
integral ( 1/ (4 (1 + u ) ) du )
If you let u = cos(4 x) then du = - 4 sin(x) dx, which doesn't match the numerator in
integral(cos( 4 x) / (1 + cos( 4 x)) dx)
so it is not possible to rearrange this integral to the same form as the first.
Had the second integral been
integral(cos( 4 x) / (1 + sin( 4 x)) dx)
then u = sin(4 x) would have yielded the same form as the first integral, and we could then say that the two are different expressions of the same problem.*@