Let `I_(n)=int_(0)^(pi//2)x^(n)cosxdx,` where in is a non-negative integer
Then `sum_(n=2)^(oo)((I_(n))/(n!)+(I_(n-2))/((n-2)!))` equals-

Options

A. `e^(pi//2)-1-(pi)/(2)`
B. `e^(pi//2)-1`
C. `e^(pi//2)-(pi)/(2)`
D. `e^(pi//2)`

Answer

A

Explanation

`I_(n)=int_(0)^(pi//2)x_(1)^(n)underset(II)cosxdx`
`=x^(n)sinx|_(0)^(pi//2)-int_(0)^(pi//2)nx^(n-1)sinxdx`
`=((pi)/(2))^(2)-0-(nx^(n-1)(-cosx)|_(0)^(pi//2)-int_(0)^(pi//2)n(n-1)x^(n-2)(-cosx)dx`
`=((pi)/(2))^(n)-0-(n-1)int_(0)^(pi//2)x^(n-2)cosxdx`
`I_(n)=((pi)/(2))^(n)-n(n-1)I_(n-2)`
`underset(n=2)overset(oo)sum((I_(n))/(n!)+(I_(n-2))/((n-2)!))=underset(n=2)overset(oo)sum((((pi)/(2))^(2)-n(n-1)I_(n-2))/(n!)+(I_(n-2))/((n-2)!))`
`underset(n=2)overset(oo)sum(((pi)/(2))^(n)(1)/(n!))`
`=((pi)/(2))^(2)(1)/(2!)+((pi)/(2))^(3)(1)/(3!)+((pi)/(2))^(4)(1)/(4!)+....`
`e^(pi//2)-1-((pi)/(2))`