pi -- 8 / | | sin(x) | ------------------- dx | 1 + sin(x) - cos(x) | / pi -- 12
Integral(sin(x)/(1 + sin(x) - cos(x)), (x, pi/12, pi/8))
Rewrite the integrand:
The integral of a constant times a function is the constant times the integral of the function:
Don't know the steps in finding this integral.
But the integral is
So, the result is:
Now simplify:
Add the constant of integration:
The answer is:
/ / 2/x\\ | log|1 + tan |-|| | sin(x) x \ \2// / /x\\ | ------------------- dx = C + - - ---------------- + log|1 + tan|-|| | 1 + sin(x) - cos(x) 2 2 \ \2// | /
/ 2\ | / ___________\ | / 2\ | | / ___ | | | / ___ ___ \ | | | / 1 \/ 2 | | | | \/ 2 \/ 6 | | | | / - + ----- | | | | ----- + ----- | | | | 1 \/ 2 4 | | | | 1 4 4 | | log|1 + |---------------- - ----------------| | log|1 + |--------------- - ---------------| | | | ___________ ___________| | / ___________\ | | ___ ___ ___ ___| | / ___ ___ \ | | / ___ / ___ | | | / ___ | | | \/ 2 \/ 6 \/ 2 \/ 6 | | | \/ 2 \/ 6 | | | / 1 \/ 2 / 1 \/ 2 | | | / 1 \/ 2 | | |- ----- + ----- - ----- + -----| | | ----- + ----- | | | / - - ----- / - - ----- | | | / - + ----- | \ \ 4 4 4 4 / / | 1 4 4 | \ \\/ 2 4 \/ 2 4 / / pi | 1 \/ 2 4 | --------------------------------------------- - log|1 + --------------- - ---------------| - ----------------------------------------------- + -- + log|1 + ---------------- - ----------------| 2 | ___ ___ ___ ___| 2 48 | ___________ ___________| | \/ 2 \/ 6 \/ 2 \/ 6 | | / ___ / ___ | | - ----- + ----- - ----- + -----| | / 1 \/ 2 / 1 \/ 2 | \ 4 4 4 4 / | / - - ----- / - - ----- | \ \/ 2 4 \/ 2 4 /
=
/ 2\ | / ___________\ | / 2\ | | / ___ | | | / ___ ___ \ | | | / 1 \/ 2 | | | | \/ 2 \/ 6 | | | | / - + ----- | | | | ----- + ----- | | | | 1 \/ 2 4 | | | | 1 4 4 | | log|1 + |---------------- - ----------------| | log|1 + |--------------- - ---------------| | | | ___________ ___________| | / ___________\ | | ___ ___ ___ ___| | / ___ ___ \ | | / ___ / ___ | | | / ___ | | | \/ 2 \/ 6 \/ 2 \/ 6 | | | \/ 2 \/ 6 | | | / 1 \/ 2 / 1 \/ 2 | | | / 1 \/ 2 | | |- ----- + ----- - ----- + -----| | | ----- + ----- | | | / - - ----- / - - ----- | | | / - + ----- | \ \ 4 4 4 4 / / | 1 4 4 | \ \\/ 2 4 \/ 2 4 / / pi | 1 \/ 2 4 | --------------------------------------------- - log|1 + --------------- - ---------------| - ----------------------------------------------- + -- + log|1 + ---------------- - ----------------| 2 | ___ ___ ___ ___| 2 48 | ___________ ___________| | \/ 2 \/ 6 \/ 2 \/ 6 | | / ___ / ___ | | - ----- + ----- - ----- + -----| | / 1 \/ 2 / 1 \/ 2 | \ 4 4 4 4 / | / - - ----- / - - ----- | \ \/ 2 4 \/ 2 4 /
log(1 + (1/(-sqrt(2)/4 + sqrt(6)/4) - (sqrt(2)/4 + sqrt(6)/4)/(-sqrt(2)/4 + sqrt(6)/4))^2)/2 - log(1 + 1/(-sqrt(2)/4 + sqrt(6)/4) - (sqrt(2)/4 + sqrt(6)/4)/(-sqrt(2)/4 + sqrt(6)/4)) - log(1 + (1/sqrt(1/2 - sqrt(2)/4) - sqrt(1/2 + sqrt(2)/4)/sqrt(1/2 - sqrt(2)/4))^2)/2 + pi/48 + log(1 + 1/sqrt(1/2 - sqrt(2)/4) - sqrt(1/2 + sqrt(2)/4)/sqrt(1/2 - sqrt(2)/4))
Use the examples entering the upper and lower limits of integration.