[Find the value of ] `int_(0)^(pi)| cos x - sin x| dx` का मान ज्ञात करे |

Explanation

` cos x - sin x = 0 rArr cos x = sin x rArr tan x = 1 = tan . Pi/4 `
` :. x = npi + pi/4 , " जहाँ " n = 0 , pm 1 , pm 2 , … ` जिसमे से केवल `pi/4,0` और `pi` के बीच है ।

अब ` I = int _(0)^(pi) |cos x - sin x | dx`
` = int _(0)^(pi//4) | cos x - sin x | dx + int_(pi//4)^(pi) |cos x - sin x| dx `
` = int _(0)^(pi//4) (cos x - sin x ) dx - int _(pi//4)^(pi) (cos x - sin x ) dx `
` = [ sin x + cos x ] _(0)^(pi//4) - [ sin x + cos x ] _(pi//4)^(pi)`
`= [ ( sin . pi/4 + cos. pi/4 ) - (sin 0 + cos 0)]`
` - [ (sin pi + cos pi ) - (sin .pi/4 + cos. pi/4)]`
` = [ (1/(sqrt(2))+1/(sqrt(2))-1)] - [ (0-1)-(1/(sqrt(2))+1/(sqrt(2)))]`
` sqrt(2)-1 - (-1 - sqrt(2)) = sqrt(2) -1 +1 + sqrt(2) = 2 sqrt(2)`