Evaluate

in this integral we will use the formula 1+tan²x=sec²x,

_{Then equation will be transform in below form-}

_{I = ∫tan}⁵x sec²x sec x dx

_{= ∫sec x tan}⁵x sec²x dx

Now put tan x=t which means sec²xdx=dt,

In this above integral put 1+t²=k²

^{that is mean tdt=kdk}

I = ∫(k⁴ + 1- 2k) k² dk

= ∫ (k⁶ + k² – 2k³)dk

Now put the value of k=(1+t²)=sec²x in above equation-