Integral and Derivative Problems

  • Thread starter rwisz
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  • #1
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Homework Statement


2 Questions here. Please help, math test tomorrow, study group failed for these 2 problems.
[tex]\int{\frac{1-sin\theta}{cos\theta}d\theta}[/tex]

and

Find [tex]\frac{dy}{dx} [/tex] if [tex]y=x^ee^x[/tex]

Homework Equations


NONE


The Attempt at a Solution


We've attempted every "u-substitution" we can think of for problem 1.

We've tried to differentiate problem 2 many times, and can't come up with the proper derivative.

thanks! Any advice is greatly appreciated.
 

Answers and Replies

  • #2
jgens
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My recomendation for the first one is to split the integral into two separate functions. Then all you have to integrate is sec(x) - tan(x) which is quite simple.

For the second problem show what work you've done as it should just be an application of the product rule.
 
  • #3
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Thanks for the advice on Number 1! We integrated successfully!

On to number 2 however, here's what we have so far:

Using simply the product rule:
[tex]y'=ex^{e-1}e^x+x^ee^x[/tex]

We've attempted to simplify down etc, but not coming up with the right answer here!
Is it perhaps using the idea a base besides e?

We're really struggling here, this is our last problem!
 
  • #4
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Why can't you find the right answer? Factor out e^x*x^(e-1) from both parts
 
  • #5
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We're both looking at your reply with the most blank stare you can ever imagine...the [tex]x^{e-1}[/tex] is not in the second term whatsoever... Am I really as stupid as you're making me feel? I'm very confused here...
 
  • #6
jgens
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As far as I can tell, the solution should be correct - aside from a restricted domain. Some thoughts on simplifications though: Factor (e^x)(x^e) out so you are left with [(e^x)(x^e)](e/x +1). That's about as simple as you can get it.
 
  • #7
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x^e = x^(e-1) * x
 
  • #8
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As far as I can tell, the solution should be correct - aside from a restricted domain. Some thoughts on simplifications though: Factor (e^x)(x^e) out so you are left with [(e^x)(x^e)](e/x +1). That's about as simple as you can get it.

I would factor out the smallest of both powers which in this case is e-1 for the power of x to get

[tex] ex^{e-1}e^x+x^ee^x = e^{x}x^{e-1}\left(e + x\right) [/tex]
 
  • #9
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We opted on calling our teacher as well. And she correctly guided us through the problem... and somehow it worked. I doubt she'll give us one like this on the test... but thanks for all your help, much obliged.

Thanks!
 
  • #10
jgens
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Either works, though I suppose your approach yields a simpler expression.
 
  • #11
HallsofIvy
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We're both looking at your reply with the most blank stare you can ever imagine...the [tex]x^{e-1}[/tex] is not in the second term whatsoever... Am I really as stupid as you're making me feel? I'm very confused here...

[itex]e^{x-1}[/itex] certainly is in the second term: the second term includes [itex]e^x[/itex] and [itex]e^x= (e^{x-1})(e)[/itex]
 
  • #12
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You mean x^(e-1)?
 
  • #13
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The first integral can even be done by simple substitution i.e
[tex]
\int{\frac{1-sin\theta}{cos\theta}d\theta}[/tex]
can be written as,
[tex]\int \frac{(sin {\frac {\theta}{2}}-~cos{\frac {\theta}{2}})^2}{(cos^2{\frac {\theta}{2}}-sin^2{\frac {\theta}{2}})}[/tex]
 
Last edited:

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