If `f(x) = sin x+cos x` and `g(x) = {:{((|x|)/(x),","x ne0),(2,","x=0):}` then the value of `int_(-pi//4)^(2pi) gof(x) dx` is equal to (a) `(3pi)/(4)` (b) `(pi)/(4)` (c) `pi` (d) None of these

Options

A. `3pi//4`
B. `pi//4`
C. `pi`
D. none of these

Answer

B

Explanation

We have,
`f(x)= sin x cos x= sqrt (2)n sin (x+(pi)/(4))`
and, ` g(x)={:{((|x|)/(x),","x ne0),(2,","x=0):}`
`rArr g(x) = {{:(1,"," x gt0),(2,","x=0),(-1,"," xlt0):}`
`:. gof (x) ={{:( 1, if x in (-pi//4"," 3pi//4) in(7pi//4","2pi)), (2 , if x = - pi/4 "," (3pi)/(4) "," (7pi)/(4)), (-1 , if x in (3pi//4 "," 7pi//4")):}`
`:. underset(-pi//4)overset(2pi)int go f(x) dx=underset(-pi//4)overset(3pi//4)int 1 dx +underset(3pi//4)overset(7pi//4)int-1dx +underset(7pi//4)overset(2pi)int 1dx`
`rArr underset(-pi//4)overset(2pi)int go f (x) dx=1((3pi)/(4)+ (pi)/(4)) -((7pi)/(4)-(3pi)/(4))+(2pi-(7pi)/(4))`
`rArr underset(-pi//4)overset(2pi)int go f (x) dx=pi -pi+(pi)/(4)=(pi)/(4)`