What is `int tan^(2) x sec^(4) x dx` equal to ?

Options

A. `(sec^(5)x)/(5) + (sec^(3) x)/(3) + c`
B. `(tan^(5)x)/(5) + (tan^(3)x)/(3) + c`
C. `(tan^(5) x)/(5) + (sec^(3)x)/(3) + c`
D. `(sec^(5)x)/(5) + (tan^(3)x)/(3) + c`

Answer

B

Explanation

Let `I = int tan^(2) x sec^(4) x dx`
Let `tan x = t`
`rArr sec^(2) x dx = dt`
`:. I = int tan^(2) x.sec^(2) x.sec^(2) x.dx`
`= int tan^(2) x (1 + tan^(2)x) sec^(2) x.dx`
`:. I = int t^(2) (1 + t^(2)) dt = int (t^(2) + t^(4)) dt`
`= (t^(5))/(5) + (t^(3))/(3) + c = (tan^(5) x)/(5) + (tan^(3)x)/(3) + c`