Question

Draw the graph of `f(x) sin^(-1)|sin x|+cos^(-1) (cos x)`. Find the range of the function. Find the points of non-differentiability. Also find the value of `int_(0)^(10pi)[sin^(-1)|sin x|+cos^(-1)(cos x)]` dx

Answer 1

`f(x)={{:(sin^(-1)(sin x)+cos^(-1)(cos x),0lexltpi),(sin^(-1)(sin x)+cos^(-1)(cos x),pilexle2pi):}`
`{{:(x+x,0lexltpi/2),((pi-x)+x,pi/2lexltpi),(-(pi-x)+2pi-x,pilexlt(3pi)/2),(-(x-2pi)+2pi-x,(3pi)/2lexle2pi):}`
`={{:(2x,0lexltpi/2),(pi,pi/2lexltpi),(pi,pilexlt(3pi)/2),(4pi-2x, (3pi)/2lexle2pi):}`

So the graph of the function for `x in [0, 2pi]`
is as shown inthe adjacent figure.
Now f(x) is a periodic function with period `2pi`, then the graph of `y = f(x)` for its domain is as follows.

From the graph, the range of the function is `[0, pi]`
Function is non-differentiable at `x=2npi" and "x=(2n+1)pi/2,n in Z`
`underset(0)overset(10pi)int[sin^(-1)|sin x|+cos^(-1)(cos x)]dx`
`=5xx("Area of trapezium formed by the x-axis and the curve in "[0, 2pi])`
`=5xx(1/2pi(2pi+pi))`
`=(15pi^(2))/(2)`