p - 6 / | | ______________ | \/ 1 - 2*sin(x) *cos(x) dx | / 0
Integral(sqrt(1 - 2*sin(x))*cos(x), (x, 0, p/6))
Let .
Then let and substitute :
The integral of a constant times a function is the constant times the integral of the function:
The integral of is when :
So, the result is:
Now substitute back in:
Add the constant of integration:
The answer is:
/ | 3/2 | ______________ (1 - 2*sin(x)) | \/ 1 - 2*sin(x) *cos(x) dx = C - ----------------- | 3 /
______________ ______________ / /p\ / /p\ /p\ / 1 - 2*sin|-| 2* / 1 - 2*sin|-| *sin|-| 1 \/ \6/ \/ \6/ \6/ - - ------------------ + --------------------------- 3 3 3
=
______________ ______________ / /p\ / /p\ /p\ / 1 - 2*sin|-| 2* / 1 - 2*sin|-| *sin|-| 1 \/ \6/ \/ \6/ \6/ - - ------------------ + --------------------------- 3 3 3
1/3 - sqrt(1 - 2*sin(p/6))/3 + 2*sqrt(1 - 2*sin(p/6))*sin(p/6)/3
Use the examples entering the upper and lower limits of integration.