Evaluate: `inttan^3x\ dx`

Explanation

`text { Let } I=int tan ^{3} x dx`

`=int tan x cdot tan ^{2} x dx `

`=int tan x(sec ^{2} x-1) d x `

`=int tan x cdot sec ^{2} x dx-int tan x dx `

`text { Putting } tan x=t text { in the Ist integral } `

`Rightarrow sec ^{2} xdx=d t `

`therefore I=int t cdot d t-int tan x dx `

`=frac{t^{2}}{2}-ln |sec x|+C `

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