`int(e^(sinx))/(cos^2x)(xcos^3x-sinx) dx`

Options

A. `(x + sec x) e^(sin x) + c`
B. `(x - sec x) e^(sin x) + c`
C. `(x + tan x) e^(sin x) + c`
D. `(x - tan x)e^(sin x) + c`

Answer

B

Explanation

Let us differentiate all the options one by one to get the expression in the question whose integral is to be found.
Here `xe^(sin x)` is the common term in all the options. So let us differentiate it first
Let `l = xe^(sin x)`
`rArr (dl)/(dx) = e^(sin x) [ x cos x + 1]`
`rArr (dl)/(dx) = (e^(sin x))/(cos^(2)x) [x cos^(3) x + cos^(2) x]`
Let `m = sec xe^(sin x)`
`rArr (dm)/(dx) = sec xe^(sin x). cos x + e^(sin x) sec x tan x`
`rArr (dm)/(dx) = e^(sin x) [1 + (sin x)/(cos^(2) x)]`
`rArr (dm)/(dx) = (e^(sin x))/(cos^(2)x) [cos^(2) x + sin x]`
Differentiation of option (a) is
`= (e^(sin x))/(cos^(2) x) [x cos^(3) x + cos^(2) x + cos^(2) x + sin x]`
`= (e^(sin x))/(cos^(2)x) [x cos^(3) x + 2 cos^(2) x + sin x]`
Differentiation of option (b) is
`= (e^(sin x))/(cos^(2) x) [x cos^(3) x + cos^(2) x - cos^(2) x - sin x]`
`= (e^(sin x))/(cos^(2)x) [x cos^(3) x - sin x]`
`:.` Option (b) is correct