Trigonometric Integral: Finding the Antiderivative of cotx + tanx

In summary, the problem involves finding the integral of the function f(x)=cotx+tanx. By manipulating the function, it can be rewritten as f(x)=\frac{2}{sin(2x)}. Using the properties (sin x)' = cos x and (ln f(x))' = f'(x) / f(x), the integral can be solved by integrating term by term. Another approach is to let u=sinx and v=cosx and use the cot x and tan x terms to integrate.
  • #1
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Homework Statement


If [itex]f(x)=cotx+tanx[/itex] then find [itex]\int{f(x)}dx[/itex]

The Attempt at a Solution


I was able to manipulate (not sure if in the right direction though) the function and resulted with [tex]f(x)=\frac{2}{sin(2x)}[/tex]

I've also tried re-arranging things in a different way, but came up with nothing useful (I think). So then I'm stuck, any ideas?
 
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  • #2
You could possibly use the fact that (sin x)' = cos x, along with the idea that (ln f(x))' = f'(x) / f(x).

Much easier to integrate term by term rather than combine the expressions into a single trig term.
 
  • #3
I'm not getting any form of f'(x)/f(x) and substituting u=sinx and du/dx=cosx doesn't seem like it gave me much either...
 
  • #4
Look at the cot x term --

cot x = cos x / sin x, which is exactly of the form f'(x) / f(x), since (sin x)' = cos x. A very similar idea applies to the tan x term.
 
  • #5
That wasn't what was suggested.

[tex]cot(x)+ tan(x)= \frac{cos(x)}{sin(x)}+ \frac{sin(x)}{cos(x)}[/tex]

Integrate by letting u= sin(x) in the first fraction and v= cos(x) in the second.
 
  • #6
Ahh thanks for all the help!

Sorry, I kept looking at my manipulated expression.
 

FAQ: Trigonometric Integral: Finding the Antiderivative of cotx + tanx

What is a trigonometric integral?

A trigonometric integral is an integral that contains trigonometric functions, such as sine, cosine, or tangent. It involves finding the area under a curve that is defined by one or more trigonometric functions.

What are some common examples of trigonometric integrals?

Some common examples of trigonometric integrals include the sine integral, cosine integral, and tangent integral. These integrals often arise in problems involving periodic functions or oscillations.

How do you solve a trigonometric integral?

To solve a trigonometric integral, you can use various integration techniques such as substitution, integration by parts, or trigonometric identities. It is important to also simplify the integral as much as possible before attempting to solve it.

What are the applications of trigonometric integrals?

Trigonometric integrals have many applications in physics, engineering, and other sciences. They are used to model and analyze periodic phenomena, such as the motion of a pendulum or the behavior of an electrical signal.

Are there any special properties of trigonometric integrals?

Yes, there are several special properties of trigonometric integrals. For example, the integral of an odd trigonometric function over a symmetric interval is always equal to zero. Also, the integral of a periodic function over one full period is equal to the area under the curve.

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