Posted by alsa on Monday, June 11, 2012 at 3:21pm.
Integral of ln(sinx+cosx) with respect to x from -pi/4 to pi/4
calculus - Count Iblis, Monday, June 11, 2012 at 3:47pm
sin(x) + cos(x) = sqrt(2) sin(x + pi/4)
So, the integral can be written as:
pi/4 ln(2) + Integral of Log[sin(x)]dx from 0 to pi/2
Let's call the integral in here I:
I = Integral of Log[sin(x)]dx from 0 to pi/2
Then note that because sin is symmetric w.r.t. reflection about x = pi/2 integrating to pi will yield twice the value:
2 I = Integral of Log[sin(x)]dx from 0 to pi
Then substitute in this integral
x = 2 y:
2 I = 2 Integral of Log[sin(2y)]dy from 0 to pi/2 ----->
I = Integral of
(Log(2) + Log[cos(y)] +Log[sin(y)] )dy from 0 to pi/2 =
pi/2 Log(2) + 2 I
because the integral of log[cos(y)] is the same as the integral of
log[sin(y)], as cos(y) is sin(pi/2-y).
Solving for I gives:
I = -pi/2 log(2)
The original integral is thus
- pi/4 log(2)
Answer This Question
More Related Questions
- Calculus - Use the symmetry of the graphs of the sine and cosine functions as an...
- Calculus - I don't know if I did these problems correctly. Can you check them? ...
- Trig........ - I need to prove that the following is true. Thanks (cosx / 1-sinx...
- Pre-Calc - Trigonometric Identities Prove: (tanx + secx -1)/(tanx - secx + 1)= ...
- Trigonometry Check - Simplify #3: [cosx-sin(90-x)sinx]/[cosx-cos(180-x)tanx] = [...
- Calculus - Evaluate the following integral integral 1 = a and b = 4 of sinx dx/(...
- Trigonometry. - ( tanx/1-cotx )+ (cotx/1-tanx)= (1+secxcscx) Good one! Generally...
- Math - Verify the identity . (cscX-cotX)^2=1-cosX/1+cosX _______ sorry i cant ...
- Calculus - If y=3/(sinx+cosx) , find dy/dx A. 3sinx-3cosx B. 3/(sinx+cosx)^2 C...
- Calculus 2 Trigonometric Substitution - I'm working this problem: ∫ [1-tan...