The value of
`int_(a)^(b) (x^(7) + sinx)/(cosx)dx` where `a + b = 0` is

Options

A. `2b - a sin (b-a)`
B. `a+3bcos (b - a)`
C. `sin a - (b-a)cos b`
D. `0`

Answer

D

Explanation

`a+b = 0 rArr a =- b`
`I = int_(-b)^(b) (x^(7) + sinx)/(cos x)`
Using property
`int_(-a)^(a) f(x) dx = {{:(int_(0)^(2a)f(x) dx," if "f(x) " is even "),(0," if "f(x) " is odd "):}`
`f(x) = (x^(7) +sin x)/(cos x)`
`f(-x) = ((-x)^(7) +sin(-x))/(cos(-x)) = (-x^(7)-sinx)/(cosx)`
`= -[(x^(7)sinx)/(cosx)]`
`= - f(x)`
So `f(x)` is odd hence
`= - f(x)`
So, `f(x)` is odd hence
`I = 0`