Question

$\sf{Find\ Integral:}\\\\\displaystyle \\\\log(sinx - cosx) \int\limits^\frac{\pi}{4}_0$

Answer 1

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Answer 2

Answer:

Using the trigonometric identities

cos x = sin (x+π/2) and sin x + sin y = 2 sin((x+y)/2) cos((x-y)/2) gives

1/(sin x + cos x) = 1/(sin x + sin(x+π/2)) = 1/(2sin(x+π/4)cos(-π/4)) = csc(x+π/4)/√2

We can integrate using the formula ∫ csc x dx = ln|tan(x/2)| + C to get

∫ dx/(sinx+cos x) = (1/√2) * ∫ [csc(x+π/4)] dx = (1/√2) ln|tan(x/2 + π/8)| + C

Using the trigonometric identities

cos x = sin (x+π/2) and sin x + sin y = 2 sin((x+y)/2) cos((x-y)/2) gives

1/(sin x + cos x) = 1/(sin x + sin(x+π/2)) = 1/(2sin(x+π/4)cos(-π/4)) = csc(x+π/4)/√2

We can integrate using the formula ∫ csc x dx = ln|tan(x/2)| + C to get

∫ dx/(sinx+cos x) = (1/√2) * ∫ [csc(x+π/4)] dx = (1/√2) ln|tan(x/2 + π/8)| + C