Trigonometric Integral, weird results

Click For Summary

Discussion Overview

The discussion revolves around the evaluation of the integral of the product of cotangent and cosecant squared functions, specifically addressing the results obtained through different substitution methods. The scope includes mathematical reasoning and exploration of integral calculus.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant presents the integral \(\int cot(x)\cdot csc^{2}(x)\cdot dx\) and finds that using the substitution \(u=cot(x)\) leads to the result \(-\frac{1}{2}cot^{2}(x)+C\).
  • The same participant also evaluates the integral \(\int \frac{cos(x)}{sin^{3}(x)}dx\) using the substitution \(u=sin(x)\) and arrives at \(-\frac{1}{2}csc^{2}(x)+C\), expressing confusion over the two different forms of the answer.
  • Another participant notes that the two answers differ only by a constant, referencing the identity \(\cot^2 x + 1 = \csc^2 x\).
  • A later reply expresses satisfaction with the resolution of the confusion, suggesting that the situation is a common experience for calculus students.

Areas of Agreement / Disagreement

Participants generally agree that the two results are equivalent up to a constant, but the initial confusion regarding the different forms of the answer remains a point of discussion.

Contextual Notes

The discussion does not delve into the specifics of the mathematical steps leading to the results, nor does it clarify the implications of the identity used to reconcile the two answers.

Yankel
Messages
390
Reaction score
0
Hello all,

I am trying to solve the integral:

\[\int cot(x)\cdot csc^{2}(x)\cdot dx\]

If I use a substitution of u=cot(x), I get

\[-\frac{1}{2}cot^{2}(x)+C\]

which is the correct answer in the book, however, if I do this:

\[\int \frac{cos(x)}{sin^{3}(x)}dx\]

I get, using a substitution u=sin(x)

\[-\frac{1}{2}csc^{2}(x)+C\]

which according to MAPLE, is also correct. In addition, according to MAPLE the move I made for the new function, is also correct !

I am confused !
 
Physics news on Phys.org
Those two answers only differ by a constant, because $\cot^2 x+ 1 = \csc^2x$.
 
Mystery solved then...I never thought of that...nice ! ;-)
 
Yankel said:
Mystery solved then...I never thought of that...nice ! ;-)

It's a "rite of passage" for calculus students. (Smile)
 

Similar threads

High School Integral of cosecant function: understanding different approaches
  • · Replies 14 ·
Replies
14
Views
4K
Undergrad Why Are There Different Forms of the Integration Formula for Cosecant?
  • · Replies 3 ·
Replies
3
Views
2K
High School Integration by parts of inverse sine, a solved exercise, some doubts...
  • · Replies 4 ·
Replies
4
Views
3K
High School Trigonometric substitution, a case I'd like to share
  • · Replies 3 ·
Replies
3
Views
3K
MHB Is the Tabular Method the Easiest Way to Solve Integrals?
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Substitution in a definite integral
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Integration Using Trigonometric Substitution
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad Different Substitution for Solving an Integral
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Did Math DF's integral calculator make a glaring mistake?
  • · Replies 8 ·
Replies
8
Views
3K
High School Integration by Parts, an introduction I get confused with
  • · Replies 2 ·
Replies
2
Views
3K