The value of
`int_(0)^(pi//4) (tan^(n)x+tan^(n-2)x)d(x-([x])/(1!)+([x]^(2))/(2!)-([x]^(3))/(3!)+....)`
where [x] is greatest function, is

Options

A. `(1)/(n)`
B. `(1)/(n+2)`
C. `(1)/(n-1)`
D. `(1)/(n-2)`

Answer

C

Explanation

For `x in[0,pi//4],[x]=0`
`:.underset(0)overset(pi//4)int (tan^(n)x+tan^(n-2)x)d(x-([x])/(1!)+([x]^(2))/(2!)-([x]^(3))/(3!)+....)`
`=underset(0)overset(pi//4)int tan^(n-2)x sec^(2)x dx=[(tan^(n-1)x)/(n-1)]_(0)^(pi//4)=(1)/(n-1)`