could anybody please explain how
sec x tan x - ¡ì sec x tan^2(x) dx
= sec x tan x + ¡ì sec x dx - ¡ì sec^3(x) dx
What I don't understand about your question is what is
¡ì ?
i just want to know if those two equations are equal, if yes, how did one go to the other.
I'm not sure what your symbols stand for either. Something didn't translate to ASCII very well when you did the paste.
I apologize for the confusion caused by the symbols in your question. The symbol ¡ì typically represents the integral symbol (∫), which denotes the process of integration in calculus.
To verify whether the two equations are equal, let's break down each step:
Given:
∫[sec(x)tan(x) - ∫[sec(x)tan^2(x)] dx
To simplify this expression, we can rewrite it as:
∫[sec(x)tan(x)] dx + ∫[sec(x)] dx - ∫[sec^3(x)] dx
Now, let's explain how the first expression transforms into the second expression:
Step 1: Looking at sec(x)tan^2(x)
We can rewrite tan^2(x) as tan(x) * tan(x), so the expression becomes:
sec(x)tan(x) - ∫[sec(x)tan(x) * tan(x)] dx
Step 2: Integration by parts
To evaluate ∫[sec(x)tan(x) * tan(x)] dx, we need to use integration by parts. The formula for integration by parts is:
∫[uv] dx = ∫[u(dv/dx)] dx - ∫[(du/dx)v] dx
Let's break down the integration by parts calculation:
u = tan(x) => du/dx = sec^2(x)
dv/dx = sec(x)tan(x) => v = ∫[sec(x)tan(x)] dx
Using the integration by parts formula, we have:
∫[sec(x)tan(x) * tan(x)] dx = tan(x) * ∫[sec(x)tan(x)] dx - ∫[(sec^2(x)) * ∫[sec(x)tan(x)] dx] dx
Step 3: Substituting the integration by parts result
Now, replacing ∫[sec(x)tan(x) * tan(x)] dx with tan(x) * ∫[sec(x)tan(x)] dx - ∫[(sec^2(x)) * ∫[sec(x)tan(x)] dx] dx in the original equation, we get:
sec(x)tan(x) - (tan(x) * ∫[sec(x)tan(x)] dx - ∫[(sec^2(x)) * ∫[sec(x)tan(x)] dx] dx)
Step 4: Simplifying the expression
Next, let's distribute the negative sign and combine like terms:
sec(x)tan(x) - tan(x) * ∫[sec(x)tan(x)] dx + ∫[(sec^2(x)) * ∫[sec(x)tan(x)] dx] dx
Step 5: Rearranging terms
Let's rearrange the terms to bring similar ones together:
sec(x)tan(x) + ∫[sec(x)] dx - ∫[sec^3(x)] dx
We have arrived at the second equation, which is:
∫[sec(x)tan(x)] dx + ∫[sec(x)] dx - ∫[sec^3(x)] dx
Therefore, we can conclude that both equations are equal.