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NastyAccident
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Homework Statement
Integral of 5x*csc(7x)*cot(7x)
Homework Equations
Integration by parts
Trigonometric Derivatives (csc(u) = -csc(u)cot(u))
Substitution
The Attempt at a Solution
*Note, I apologize beforehand for the lack of knowledge on how to display mathematical equations nicely on PF *
5*int[x*csc(7x)*cot(7x)]
I am preparing for the 7x substitution here:
5/7*int[7x*csc(7x)*cot(7x)]
let z = 7x
dz = 7dx
5/7 * 1/7 *int[z*csc(z)*cot(z)]
5/49 * int[z*csc(z)*cot(z)]
Preparing for the csc(z) substitution here:
-1* 5/49 * int[-1*z*csc(z)*cot(z)]
let w = csc(z)
dw = -csc(z)cot(z)
arccsc(w) = z
-1*5/49*int[arccsc(w)]
Integration by parts
u = arccsc(w) dv=dw
du = -1/|w|*sqrt(w^2-1) v = w
-1*5/49*[w*arccsc(w) - int[ -1/|w|*sqrt(w^2-1) * w ] ]
-1*5/49*[w*arccsc(w) + int[w/|w|*sqrt(w^2-1)]
In between these two steps I may have made an error since I canceled the absolute value w on the denominator with the w on the numerator.
-1*5/49*[w*arccsc(w) + int[1/sqrt(w^2-1)]
Solved for answer in two different ways (both wrong)
arccosh way -
-1*5/49*[w*arccsc(w) + arccosh(w)]
-1*5/49*[csc(z)*arccsc(csc(z)) + arccosh(csc(z))]
-1*5/49*[csc(7x)*arccsc(csc(7x)) + arccosh(csc(7x))]
trig substitution way -
-1*5/49*[w*arccsc(w) + int[1/sqrt(w^2-1)]
let w = 1*sec(theta)
dw = sec(theta)tan(theta)
-1*5/49*[w*arccsc(w) + int[sec(theta)tan(theta)/sqrt(sec(theta)^2-1)]
-1*5/49*[w*arccsc(w) + int[sec(theta)tan(theta)/tan(theta)]
-1*5/49*[w*arccsc(w) + int[sec(theta)]
-1*5/49*[w*arccsc(w) + ln|sec(theta)+tan(theta)|]
Building the right triangle
Since w/1 = sec(theta)
w is the hyp, 1 is the adj side, & sqrt(w^2-1) is then the opposite side
w -> /| <- sqrt(w^2-1)
/ |
---
^ (1)
tan(theta) = sqrt(w^2-1)
sec(theta) = w
-1*5/49*[w*arccsc(w) + ln|w+sqrt(w^2-1)|]
Final answer:
-1*5/49*[csc(7x)*arccsc(csc(7x)) + ln|csc(7x)+sqrt(csc(7x)^2-1)|]
So, I'm unsure as to why both of these answers were marked wrong (I've submitted them into webassign). Any help would be appreciated!
http://img168.imageshack.us/img168/3822/mathwoesonwebassign5xcs.th.jpg
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