Question

Calculus Help!!
Is there a faster way to do this?
∫csc^10tcot^3tdt

Answer 1

Faster than what?

Answer 2

http://www.wolframalpha.com/input/?i=indefinite+integral&a=*C.indefinite+integral-_*Calculator.dflt-&f2=%28cscx^10%29%28cotx^3%29&x=0&y=0&f=Integral.integrand_%28cscx^10%29%28cotx^3%29&a=*FVarOpt.1-_**-.***Integral.rangestart-.*Integral.rangeend--.**Integral.variable---.*--

Answer 3

XD that doesnt look good

Answer 4

is it we can put u=csct and du=-csctcotdt?
which is faster?

Answer 5

No, I do not believe you can do u substitution in that case because of the powers on them.

Answer 6

\[\int\csc^{10}t\cot^3tdt=\int\csc^{9}t\cot^2t(\csc t\cot t)dt\]\[=\int\csc^9t(1-\csc^2t)(\csc t\cot t)dt=\int\csc^9t-\csc^{11}t(\csc t\cot t)dt\]\[u=\csc t\]\[du=-\csc t\cot tdt\]

Answer 7

whoops, second part is backwards...

Answer 8

should have been\[\int\csc^9t(\csc^2t-1)(\csc t\cot t)dt=\int\csc^{11}t-\csc^9t(\csc t\cot t)dt\]now do the u sub above

Answer 9

forgot parentheses above...\[\int\csc^9t(\csc^2t-1)(\csc t\cot t)dt=\int(\csc^{11}t-\csc^9t)(\csc t\cot t)dt\]

Answer 10

thank you so much!!

Answer 11

when do we have to use the reduction formulas?

Answer 12

when all else fails
I hate them personally, and almost always find a way around them
sometimes I wind up integrating these types of things by parts (which is the same as the reduction formula) and don't even notice

Answer 13

Thank you!!