1 / | | sin(2*x)*sin(x) | --------------- dx | 1 + cos(x) | / 0
Integral((sin(2*x)*sin(x))/(1 + cos(x)), (x, 0, 1))
There are multiple ways to do this integral.
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:
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:
/ /x\ 3/x\ 2/x\ 4/x\ | 4*tan|-| 12*tan |-| 4*x*tan |-| 2*x*tan |-| | sin(2*x)*sin(x) 2*x \2/ \2/ \2/ \2/ | --------------- dx = C - ------------------------- + ------------------------- + ------------------------- - ------------------------- - ------------------------- | 1 + cos(x) 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ | 2 + 2*tan |-| + 4*tan |-| 2 + 2*tan |-| + 4*tan |-| 2 + 2*tan |-| + 4*tan |-| 2 + 2*tan |-| + 4*tan |-| 2 + 2*tan |-| + 4*tan |-| / \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/
2 4 3 2 4*tan (1/2) 2*tan (1/2) 4*tan(1/2) 12*tan (1/2) - ----------------------------- - ----------------------------- - ----------------------------- + ----------------------------- + ----------------------------- 4 2 4 2 4 2 4 2 4 2 2 + 2*tan (1/2) + 4*tan (1/2) 2 + 2*tan (1/2) + 4*tan (1/2) 2 + 2*tan (1/2) + 4*tan (1/2) 2 + 2*tan (1/2) + 4*tan (1/2) 2 + 2*tan (1/2) + 4*tan (1/2)
=
2 4 3 2 4*tan (1/2) 2*tan (1/2) 4*tan(1/2) 12*tan (1/2) - ----------------------------- - ----------------------------- - ----------------------------- + ----------------------------- + ----------------------------- 4 2 4 2 4 2 4 2 4 2 2 + 2*tan (1/2) + 4*tan (1/2) 2 + 2*tan (1/2) + 4*tan (1/2) 2 + 2*tan (1/2) + 4*tan (1/2) 2 + 2*tan (1/2) + 4*tan (1/2) 2 + 2*tan (1/2) + 4*tan (1/2)
-2/(2 + 2*tan(1/2)^4 + 4*tan(1/2)^2) - 4*tan(1/2)^2/(2 + 2*tan(1/2)^4 + 4*tan(1/2)^2) - 2*tan(1/2)^4/(2 + 2*tan(1/2)^4 + 4*tan(1/2)^2) + 4*tan(1/2)/(2 + 2*tan(1/2)^4 + 4*tan(1/2)^2) + 12*tan(1/2)^3/(2 + 2*tan(1/2)^4 + 4*tan(1/2)^2)
Use the examples entering the upper and lower limits of integration.