Posted by alsa on Monday, June 11, 2012 at 3:21pm.
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)
Related Questions
Calculus - Use the symmetry of the graphs of the sine and cosine functions as an...
calculus - Find an equation of the tangent line to the curve at y = cosx - sinx...
Calculus - Evaluate the integral: 16csc(x) dx from pi/2 to pi (and determine if ...
CALC - 1. The problem statement, all variables and given/known data cos(x) = -2 ...
Calc - 1. The problem statement, all variables and given/known data cos(x) = -2 ...
Math(Please help) - 2) Use the sum and difference identites sin[x + pi/4] + sin[...
Math - Solve (sinx-1)(cosx -1/2) = 0 where 0≤x<2pi a) pi/3, pi/...
Calculus II - How do you integrate the function -- > sinx (4^cosx) from 0...
Math(Please help) - 1)tan Q = -3/4 Find cosQ -3^2 + 4^2 = x^2 9+16 = sqrt 25 = 5...
Math - Calculus - Integration of (sinx + cosx)^2 from 0 to pi/2. I get pi/2 + 1 ...
For Further Reading
