Trigonometric Integral, weird results

In summary, the conversation discusses solving two integrals using different substitutions and how the resulting answers are different by only a constant. The reason for this is because $\cot^2x+1=\csc^2x$. The conversation concludes that this is a common occurrence for calculus students.
  • #1
Yankel
395
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 !
 
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  • #2
Those two answers only differ by a constant, because $\cot^2 x+ 1 = \csc^2x$.
 
  • #3
Mystery solved then...I never thought of that...nice ! ;-)
 
  • #4
Yankel said:
Mystery solved then...I never thought of that...nice ! ;-)

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

1. What is a trigonometric integral?

A trigonometric integral is a mathematical expression that involves trigonometric functions such as sine, cosine, tangent, etc. and integration. It represents the area under a curve of a trigonometric function.

2. Why do trigonometric integrals sometimes give weird results?

Trigonometric integrals can give weird results due to the complex nature of trigonometric functions. These functions can have multiple solutions and can also produce infinite series. Therefore, when integrating these functions, the result can be unexpected or unconventional.

3. Can weird results from trigonometric integrals be correct?

Yes, even though the results may seem strange or unusual, they can still be mathematically correct. These results may involve complex numbers or irrational numbers, which can be valid solutions to trigonometric integrals.

4. How can I check the accuracy of the result from a trigonometric integral?

You can check the accuracy of the result by using a graphing calculator or software to plot the original function and the integral function. The area under the curve should match the result of the integral. Additionally, you can also solve the integral using different methods to ensure the accuracy of the result.

5. Are there any real-world applications of trigonometric integrals with weird results?

Yes, trigonometric integrals are used in physics, engineering, and other fields to solve problems involving periodic functions. The weird results can represent physical phenomena such as oscillations, vibrations, and waves. These results can also be seen in the analysis of electrical signals and sound waves.

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