How to integrate tan x/ (1 - cos x) ?

In summary, to simplify tan x/ (1 - cos x), you can use the trigonometric identity 1 - cos x = sin^2 x and then apply the quotient identity for tangent. The domain of this expression is all real numbers except for values of x that make the denominator equal to 0. To integrate tan x/ (1 - cos x), you can use the substitution method and then solve using the power rule of integration. The derivative of tan x/ (1 - cos x) can be found using the quotient rule of differentiation. Finally, the graph of tan x/ (1 - cos x) is not a simple curve and can be plotted using a graphing calculator or software.
  • #1
robierob12
48
0
Im doing a differential equation and end up with my last step, which is an integration.

I can't seem to remember how to integrate this one


int. (tanx)/(1-cosx)

hmmm

[(sinx)/(cosx)] / (1-cosx)

u = cosx
du = -sinx dx

throw a negative outside the integral.


du/(u)(1-u)

maby? then use some easy partial frac. decomp.

would that be a correct method to use or is there another way.


Rob
 
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  • #2
Partial fractions is the method I'd use.
 

1. How do you simplify tan x/ (1 - cos x)?

To simplify tan x/ (1 - cos x), you can use the trigonometric identity 1 - cos x = sin^2 x. This will result in the simplified expression of tan x/ sin^2 x. From there, you can use the quotient identity for tangent to further simplify the expression.

2. What is the domain of tan x/ (1 - cos x)?

The domain of tan x/ (1 - cos x) is all real numbers except for values of x that make the denominator equal to 0. In this case, x cannot equal π/2 + nπ, where n is any integer.

3. How do you integrate tan x/ (1 - cos x)?

To integrate tan x/ (1 - cos x), you can use the substitution method. Let u = cos x, then du = -sin x dx. This will result in the integral becoming -∫1/u du, which can be easily solved using the power rule of integration.

4. What is the derivative of tan x/ (1 - cos x)?

The derivative of tan x/ (1 - cos x) is (1 + sin x)/ (1 - cos x)^2. You can use the quotient rule of differentiation to find this derivative.

5. What is the graph of tan x/ (1 - cos x)?

The graph of tan x/ (1 - cos x) is not a simple curve, as it involves both tangent and cosine functions. However, you can use a graphing calculator or software to plot the graph and see its shape and behavior.

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