Evaluate : (i) `inttanxdx` (ii) `intcotxdx`
(iii) `intsecxdx` (iv) `int"cosec x "dx`

Explanation

(i) `inttanxdx=int(sinx)/(cosx)dx`
`=-intdt,"where"cosx=t andsindx=-dt`
`=-log|t|+C=-log|cosx|+C`.
`:.int tanx dx=-log|cosx|+C`.
(ii) `intcotxdx=int(cosx)/(sinx)dx`
`=int(1)/(t)dt," where"sinx=t andcosxdx=dt`
`=log|t|+C=log|sinx|+C`.
`:.intcotxdx=log|sinx|+C`.
(iii) `intsecdx=int(secx(secx+tan+tanx))/((secx+tanx))dx`
[multilplying numerator and denominator by (sec x+tanx)]
`=int(1)/(t)"dt, where "(secx+tanx)=t`
`and sec x (sec x + tan x)dx=dt`
`=log|t|+C=log|{:(secx+tanx):}|+C`.
`:.intsecxdc=log|{:(secx+tanx):}|+C`.
(iv) `int"cosec x dx "=int("cosec "x("cosec "x=cotx))/(("cosec "x-cotx))dx`
[multiplying numerator and denominator by (cosec x - cot x)]
`=int(1)/(t)"dt, where "("cosec "x-cotx)=t`
and cosec x (cosec x-cot x)dx=dt
`=log|t|+C=log|{:"cosec "x-cotx:}|+C`.
`:.int"cosec "xdx=log|{:"cosec "x-cotx:}|+C`.