Evaluating the Integral of csc3 x * tan3 x
In this article, we will evaluate the integral I ∫ csc3 x * tan3 x dx. This involves multiple steps and techniques including conversion, substitution, and integration by parts. Let's break down the process step-by-step.
Step-by-Step Solution
Given the integral:
Step 1: Conversion and Substitution
First, we convert the integral into terms of sine and cosine:
Let's start with:
B ∫ csc3 x * tan3 x dx
Recall that:
csc x 1/sin x and tan x sin x / cos x
Thus, we can rewrite B as:
B ∫ (1/sin3 x) * (sin2 x / cos3 x) dx
This simplifies to:
B ∫ sec3 x dx
Step 2: Integration by Parts
To evaluate the integral of sec3 x, we use integration by parts. Let:
u sec x and dv sec2 x dx
Then:
du sec x * tan x dx and v tan x
Using the integration by parts formula:
I uv - ∫ vdu
We get:
I sec x * tan x - ∫ tan x * sec x * tan x dx
This can be further simplified to:
I sec x * tan x - ∫ sec3 x dx ∫ sec x dx
Let I represent the integral of sec3 x. Thus, we have:
I sec x * tan x - I ∫ sec x dx
Multiplying both sides by 2:
2I sec x * tan x - ∫ sec x dx
Now, we need to evaluate ∫ sec x dx. The standard integral for this is:
∫ sec x dx ln |sec x tan x| C
So, substituting back:
2I sec x * tan x - ln |sec x tan x| C
Finally, dividing by 2:
I (sec x * tan x - ln |sec x tan x|) / 2 C
Final Answer
Therefore, the integral ∫ csc3 x * tan3 x dx evaluates to:
I (sec x * tan x - ln |sec x tan x|) / 2 C
Conclusion
By carefully converting the integral, using substitution, and applying integration by parts, we were able to solve for the given integral. Understanding these techniques is crucial for evaluating more complex integrals in calculus.
For further exercises and practice, try evaluating similar integrals or explore advanced calculus topics to strengthen your skills.