anonymous
  • anonymous
\[\int\cos^2u\left(\sin u\right)^{-1}\,du\]
Mathematics
  • Stacey Warren - Expert brainly.com
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schrodinger
  • schrodinger
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experimentX
  • experimentX
1/sinu+sinu^2/sinu
anonymous
  • anonymous
What?
experimentX
  • experimentX
expand cos ... separate denominator, you will get int(cscx) + int (sinx)

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experimentX
  • experimentX
oops sorry - at middle
Callisto
  • Callisto
(cos)^2 = 1-(sin)^2 (cos)^2/ sin = [1-(sin)^2] / sin = csc - sin ..
anonymous
  • anonymous
Oh, okay; whooops. That makes sense
.Sam.
  • .Sam.
\[\begin{array}{l} \text{For the integrand }\cos (u) \cot (u)\text{, write }\cos (u) \cot (u)\text{ as }\csc (u)-\sin (u): \\ \text{}=\int\limits (\csc (u)-\sin (u)) \, du \\ \text{Integrate the sum term by term and factor out constants:} \\ \text{}=\int\limits \csc (u) \, du-\int\limits \sin (u) \, du \\ \text{The integral limits of }\sin (u)\text{ is }-\cos (u): \\ \text{}=\cos (u)+\int\limits \csc (u) \, du \\ \text{The integral limits of }\csc (u)\text{ is }-\log (\cot (u)+\csc (u)): \\ \text{}=\cos (u)-\ln (\cot (u)+\csc (u))+\text{constant} \\ \text{Which is equivalent for restricted }u\text{ values \to:} \\ \text{}=\cos (u)+\ln \left(\sin \left(\frac{u}{2}\right)\right)-\ln \left(\cos \left(\frac{u}{2}\right)\right)+\text{constant} \\\end{array}\]
anonymous
  • anonymous
Wait, how do I prove \(\int\csc u\,du=-\log\left(\cot u+\csc u\right)\)?
experimentX
  • experimentX
differentiate the left hand side .. it's pretty obvious, you can also find the steps
anonymous
  • anonymous
I don't follow.
Callisto
  • Callisto
see the attachment
1 Attachment
anonymous
  • anonymous
Any other tricks like that I should know? Like multiplying top and bottom by some trig identity that allows me to evaluate it?
Callisto
  • Callisto
similar for secx
Callisto
  • Callisto
tan -> integration by substitution, similar for cot

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