Evaluate the following integrals :
`(i) int tan^(-5)x sec^(6)xdx` `(ii) intcosec^(8)xcot^(3)xdx`

Explanation

`(i)` We have,
`intan^(-5)xsec^(6)xdx`
`=int(sec^(6)x)/(tan^(5)x)dx=int((1+tan^(2)x)^(2))/(tan^(5)x)sec^(2)xdx`
`=int(1+2u^(2)+u^(4))/(u^(5))du`, `tanx=uimpliessec^(2)x=du`
`=int((1)/(u^(5))+(2)/(u^(3))+(1)/(u))du`
`=-(1)/(4u^(4))-(2)/(2u^(2))+ln|u|+C`
`=-(1)/(4)cot^(4)x-cot^(2)x+ln|tanx|+C`, (`C` is a constant of integration).
`(ii)` We have
`intcosec^(8)xcotx^(3)xdx=int(1+cot^(2)x)^(3)cot^(3)xcosec^(2)xdx`
`=-int(1+u^(6)+3u^(2)+3u^(4))u^(3)du` Let us put `cotx=uimpliescosec^(2)xdx=-du`
`=-[(u^(10))/(10)+(3u^(8))/(8)+(3u^(6))/(6)+(u^(4))/(4)]+C`
`=-[(cot^(10)x)/(10)+(3cot^(8)x)/(8)+(cot^(6)x)/(2)+(cot^(4)x)/(4)]+C`, (`C` is constant)
`(iii)` We have
`intcot^(4)xcosec^(6)xdx`
`=intcot^(4)x(cot^(2)x+1)^(2)*cosec^(2)xdx`
`=intcot^(4)x(1+cot^(4)x+2cot^(2)x)cosec^(2)xdx` Let us put `cotx=u`, `cosec^(2)xdx=-du`
`=intu^(4)(1+u^(4)+2u^(2))*(-du)`
`=-[(u^(9))/(9)+(2u^(7))/(7)+(u^(5))/(5)]+C`
`=-[(cot^(9)x)/(9)+(2cot^(7)x)/(7)+(cot^(5)x)/(5)]+C`, (`C` is a constant of integration).