If the value `int(1-(cotx)^(2008))/(tanx+(cotx)^(2009))dx=1/k ln|sin^kx+cos^kx|+c`, then find `k.`

Answer

2010

Explanation

` I=int(1-cot^(2008)x)/(tanx+cot^(2009)x)dx`
`=int(sin^(2008)x-cos^(2008)x)/(sin^(2008)x[(sinx)/(cosx)+(cos^(2009)x)/(sin^(2009)x)])dx`
`=int(sinx cosx[sin^(2008)x-cos^(2008)x])/(sin^(2010)x+cos^(2010)x)dx`
`=int(sin^(2009)xcosx-cos^(2009)xsinx)/(sin^(2010)x+cos^(2010)x)dx`
`"Put " sin^(2010)x+cos^(2010)x=t.`
` :. I=(1)/(2010)int(dt)/(t) `
`=(1)/(2010)log|sin^(2010)x+cos^(2010)x|+C`
` :. k=2010`