Find the indefinite integral of 1-tanx/1+tanx
Dont know how to really approach this question. Should i use identities, or is there a power series i can use?
There are standard formulae for integrals of rational functions of trigonometric formulae. In this case, you can simplify things as follows.
Let's use the abbreviation:
t = tan(x)
s = sin(x)
c = cos(x)
We can write:
(1-t)/(1+t) =
(c-s)/(c+s) =
(c - s)^2/(c^2 - s^2) =
(c^2 + s^2 - 2cs)/(c^2 - s^2)
Then use the trigonometric identities:
c^2 + s^2 = 1
2 cs = sin(2x)
c^2 - s^2 = cos(2x)
to obtain:
(1-t)/(1+t) =
1/(cos(2x)) - tan(2x)
Integrating tan(2x) is trivial. You can integrate 1/cos(2x) e.g. by putting
x = (pi/4 - u), so that cos(2x) =
sin(2u. Then
1/sin(2u) =
[cos^2(u) + sin^2(u)]/[2 sin(u)cos(u)] =
1/2 [cot(u) + tan(u)]
which is trivial to integrate.
how did you get from (c-s)/(c+s) to
(c - s)^2/(c^2 - s^2)?
Multiply numerator and denominator by (c-s).
thanks
To find the indefinite integral of 1 - tan(x) / (1 + tan(x)), we can simplify the expression using trigonometric identities before integrating.
First, let's use the identity tan(x) = sin(x) / cos(x) to rewrite the integrand:
1 - tan(x) / (1 + tan(x))
= 1 - (sin(x) / cos(x)) / (1 + sin(x) / cos(x))
= 1 - (sin(x) / cos(x)) / ((cos(x) + sin(x)) / cos(x))
= 1 - sin(x) / (cos(x) + sin(x))
= (cos(x) + sin(x) - sin(x)) / (cos(x) + sin(x))
= cos(x) / (cos(x) + sin(x))
Now, we can proceed with integrating cos(x) / (cos(x) + sin(x)). One way to go about this is to use a substitution.
Let u = cos(x) + sin(x). Then, differentiate both sides with respect to x:
du/dx = -sin(x) + cos(x)
Rearranging the equation, we have:
du = (-sin(x) + cos(x)) dx
Now, we can substitute these variables in the integral:
∫ (cos(x) / (cos(x) + sin(x))) dx
= ∫ (1/u) du
After making the substitution, the integral becomes much simpler:
∫ (1/u) du = ln|u| + C
Substituting back u = cos(x) + sin(x):
∫ (1 - tan(x) / (1 + tan(x))) dx = ln|cos(x) + sin(x)| + C
So, the indefinite integral of 1 - tan(x) / (1 + tan(x)) is ln|cos(x) + sin(x)| + C, where C is the constant of integration.