Thursday, February 10, 2011
Wednesday, February 9, 2011
FINALLY
Finally, after hours of work, I have figured out the secret to trigonometric substitutions.
The biggest tip I can give is to restrict the domain of your sinθ, secθ, and tanθ functions.
For sinθ -π/2 < θ < π/2
secθ 0 < θ < π/2 OR π < θ < 3π/2
tanθ -π/2 < θ < π/2
NOTE: Be cautious when it comes to reference triangles. Most of these will only need one reference triangle due to the restriction of the domain.
The biggest tip I can give is to restrict the domain of your sinθ, secθ, and tanθ functions.
For sinθ -π/2 < θ < π/2
secθ 0 < θ < π/2 OR π < θ < 3π/2
tanθ -π/2 < θ < π/2
NOTE: Be cautious when it comes to reference triangles. Most of these will only need one reference triangle due to the restriction of the domain.
Before I start to go over what I've been doing in regards to trigonometric substitution, it'd be helpful to explain integration by parts.
Integrating by parts is based off the product rule and can be summed up like this:
where int = an integral.
int( u dv) = uv - int( v du).
Essentially you break down the initial integral into the parts u and dv with dv being the most complicated function that you are able to easily integrate.
It's best to set it up like so:
u = ???? dv = ????
du = ???? v = ????
If you maintain this order, it's easy to integrate regardless of what letters you are using for u and v.
Integrating by parts is based off the product rule and can be summed up like this:
where int = an integral.
int( u dv) = uv - int( v du).
Essentially you break down the initial integral into the parts u and dv with dv being the most complicated function that you are able to easily integrate.
It's best to set it up like so:
u = ???? dv = ????
du = ???? v = ????
If you maintain this order, it's easy to integrate regardless of what letters you are using for u and v.
I'm having issues remembering the domains and ranges of the inverse trigonometric functions, so I think it would be helpful to anyone reading that I post them here.
arccosx Domain: -1 < x < 1
Range: 0 < y < π
arcsinx Domain: -1 < x < 1
Range: -π/2 < y < π/2
arctanx Domain: all values of x
Range: -π/2 < y < π/2
arcsecx Domain: all values of x except -1 < x < 1
Range: 0 < y < π arcsecx =/= π/2
arccscx Domain: The same as arcsecx
Range: -π/2 < y < π/2
arccotx Domain: All values of x
Range: -π/2 < y < π/2
I've got copious amounts to do tonight, so I'll probably be posting random tidbits all evening.
arccosx Domain: -1 < x < 1
Range: 0 < y < π
arcsinx Domain: -1 < x < 1
Range: -π/2 < y < π/2
arctanx Domain: all values of x
Range: -π/2 < y < π/2
arcsecx Domain: all values of x except -1 < x < 1
Range: 0 < y < π arcsecx =/= π/2
arccscx Domain: The same as arcsecx
Range: -π/2 < y < π/2
arccotx Domain: All values of x
Range: -π/2 < y < π/2
I've got copious amounts to do tonight, so I'll probably be posting random tidbits all evening.
Calculus II
Hello all! This is my blog pertaining to what I'm learning in Calc II currently and what I'm confused with and working on.
Currently, we are working on trigonometric substitution for integrals.
I will post more after I finish my homework!
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