If `f(x)= int_0^(sinx) cos^(-1)t dt +int_(0)^(cosx) sin^(-1)t dt, 0 lt x lt (pi)/(2)` then ` f(pi//4)` is equal to

Options

A. `(pi)/(sqrt(2))`
B. `1+(pi)/(2sqrt(2))`
C. 1
D. none of these

Answer

B

Explanation

We have,
`f(x)= underset(0)overset(sin x) int cos^(-1)t dt +underset(0)overset(cos x) int sin^(-1)t dt`
`rArr f(x)= underset(0)overset(sin x) int ((pi)/(2)-sin^(-1)) dt+overset(cosx)underset(0)int ((pi)/(2)-cos^(-1))dt`
`rArr f(x)=(pi)/(2)sin x +(pi)/(2) cos x- underset(0)overset(sin x) int sin^(-1)t dt -underset(0)overset(cos x) int cos^(-1)t dt`
`rArr f(x)=(pi)/(2)(sinx +cos x)-[ underset(0)overset(sin x) int sin^(-1)t dt -underset(0)overset(cos x) int cos^(-1)t dt]`
Now, `underset(0)overset(sinx)intsin^(-1) t dt =underset(0)overset(x)int theta cos theta d theta, "where" t= sin theta`
`rArr underset(0)overset(sinx)intsin^(-1) t dt =[theta sin theta+cos theta] _(0)^(x)= x sin x+cos x-1` and ,
`rArr underset(0)overset(cosx)int cos^(-1) t dt=-underset(0)overset(x)int alpha sin d, alpha "where" t = cos alpha`
`rArr underset(0)overset(cosx)int cos^(-1) t dt=-[- alpha cos alpha + sin alpha] _(0)^(x)`
`rArr underset(0)overset(cosx)int cos^(-1) t dt =x cos x -sin x`
`:. f(x)=(pi)/(2) (sin x +cos x) -[ x sin x+cos x-1+ x cos x -sin x]`
`rArr f((pi)/(4))=(pi)/(2)((2)/(sqrt(2)))-[(pi)/(4sqrt(2))+(1)/(sqrt(2))-1+(pi)/(4sqrt(2))-(1)/(sqrt(2))]`
`rArr f((pi)/(4))=(pi)/(sqrt(2))-(pi)/(2sqrt(2))+1=(pi)/(2sqrt(2))+1`