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Homework Statement
Evaluate
integral dx/sqrt(x^249)
Homework Equations
The Attempt at a Solution
SEE POST #4 BELOW WITH ATTACHMENT TO VIEW MY SOLUTION EASIER TO READ
i was trying to use a csc(theta) substitution and I don't see how my logic and math below is flawed but apparently I'm getting a different answer, note that I wanted to try and evaluate this integral this way and not other ways and don't see how evaluating it this way is wrong as long as my intermediate steps are mathematically correct
integral dx/sqrt(x^249)
sense csc(theta)= hypotenuse/adjacent
let hypotenuse = x
and adjacent = 7
use the fact that hypotenuse= adjacent^2 + opposite^2
then opposite = sqrt(x^249)
then csc(theta)=x/7
then x = 7csc(theta)
use the fact that d/dx csc(theta) = cot(theta)csc(theta)
dx/dtheta = 7cot(theta)csc(theta)
dx=7cot(theta)csc(theta)dtheta
substitute dx and x back into the integral
integral[ (7cot(theta)csc(theta)dtheta)/sqrt((7csc(theta))^249) ]
took out the constants and squared the term
7*integral[ (cot(theta)csc(theta)dtheta)/sqrt(49csc(theta)^249) ]
factored out 49
7*integral[ (cot(theta)csc(theta)dtheta)/sqrt(49(csc(theta)^21)) ]
use fact that cot(theta)^2=csc(theta)^21
7*integral[ (cot(theta)csc(theta)dtheta)/sqrt(49cot(theta)^2) ]
evaluated the square root
7*integral[ (cot(theta)csc(theta)dtheta)/(7cot(theta)) ]
"canceled" cot(theta)/cot(theta)=1 and took out constant and got 7/7=1
 integral csc(theta)dtheta
used the fact that integral csc(theta)dtheta = ln(cot(theta)+csc(theta))
*ln(cot(theta)+csc(theta)) + c
it became positive
ln(cot(theta)+csc(theta)) + c
already established that csc(theta)= x/7
ln(cot(theta)+x/7) + c
use fact that cot(theta)=adjacent/opposite = 7/sqrt(x^249)
ln( 7/sqrt(x^249) + x/7 ) + c
so this gives me the wrong answer the correct one is
ln(sqrt(x^249)/7 + x/7 ) + c
but i really don't see how what i did is wrong...
SEE POST #4 BELOW WITH ATTACHMENT TO VIEW MY SOLUTION EASIER TO READ
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