`tan^(5)x`

Text Solution

Verified by Experts

`"Let I"=int tan^(5)xdx=inttan^(3)x.tan^(2)dx`
`=inttan^(3)x(sec^(2)x-1)dx`
`=int(tan^(3)xsec^(2)x-tan^(3)x)dx`
`=inttan^(3)x.sec^(2)xdx-inttanx.tan^(2)xdx`
`=inttan^(3)x.sec^(2)xdx-int tanx(sec^(2)x-1)dx`
`=inttan^(3)x.sec^(2)xdx-int tanx(sec^(2)x-tan x)dx`
`=inttan^(3)x.sec^(2)xdx-int tanxsec^(2)x dx+int tan xdx`
`=inttan^(3)x.sec^(2)xdx-int tanxsec^(2)x" "dx+log|secx|`
Put tan x=t. Then `sec^(2)x dx=dt`
`therefore I=int t^(3)dt-int t dt+log|secx|`
`=(t^(4))/(4)-(t^(2))/(2)+log|sec x|+c`
`=(tan^(4x))/(4)-(tan^(2)x)/(2)+log|sec x|+c`
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