Trigonometric Integration: Understanding the Magic Behind Integration Techniques

In summary, the conversation was about an integration problem involving csc(theta) and the use of a clever trick to simplify the integrand. The trick involved multiplying the integrand by 1 and using a substitution to make the integral easier to solve. It was also mentioned that a similar technique is used for integrating sec(theta).
  • #1
mateomy
307
0
This is more of a trig question but its coming out of an integration problem I am dealing with...


the initial problem is...

[tex]
\int\frac{\sqrt(1+x^2)}{x}dx
[/tex]

I've worked through the problem with the book and down the line I have to integrate csc(theta) without using a handy-dandy table of integrals. Anyway, the book shows the following...

[tex]
\int csc\theta \frac{csc\theta - cot\theta}{csc\theta - cot\theta} d\theta
[/tex]

I can't justify to myself why they did that, let alone what magical hat they pulled that from. Can someone show me the light? Thanks.
 
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  • #2
What is the derivative of the integrand's denominator?

What is the integrand's numerator after multiplying through by cscθ ?
 
  • #3
mateomy said:
This is more of a trig question but its coming out of an integration problem I am dealing with...


the initial problem is...

[tex]
\int\frac{\sqrt(1+x^2)}{x}dx
[/tex]

I've worked through the problem with the book and down the line I have to integrate csc(theta) without using a handy-dandy table of integrals. Anyway, the book shows the following...

[tex]
\int csc\theta \frac{csc\theta - cot\theta}{csc\theta - cot\theta} d\theta
[/tex]

I can't justify to myself why they did that, let alone what magical hat they pulled that from. Can someone show me the light? Thanks.
What they've done is to multiply the integrand by 1, which you can always do, and which doesn't change the value of the integrand.


The reason they did this is that the numerator is [csc2(theta) - csc(theta)cot(theta)] d(theta), which just happens to be the differential of the denominator, which is -cot(theta) + csc(theta).

You can make the substitution u = -cot(theta) + csc(theta), so du = csc2(theta) - csc(theta)cot(theta). This leads to an easy integral.
 
  • #4
Okay, I can see it. I also understand what you're saying by multiplying by 1. My problem is just HOW they got that particular equivalent of 1? Was it just being clever? That's not an identity right?

Thanks, btw.
 
  • #5
I mean...just the simple {csc + cot}, not the whole rational.
 
  • #6
I think it was a clever trick that someone discovered. A similar technique is used to integrate sec(theta).
 
  • #7
Oh okay, (sigh of relief)...I thought my mind was gone.
 

1. What is trigonometric integration?

Trigonometric integration is a method used in calculus to solve integrals that involve trigonometric functions, such as sine, cosine, and tangent. It involves using various trigonometric identities and substitution techniques to simplify the integral and find its solution.

2. Why is trigonometric integration important?

Trigonometric integration is important because it allows us to solve a wide variety of integrals that would otherwise be difficult or impossible to solve. It is also essential in many areas of science, engineering, and mathematics, as it helps us understand and model real-world phenomena.

3. What are some common trigonometric identities used in integration?

Some common trigonometric identities used in integration include the Pythagorean identities, double angle identities, and power-reducing identities. These identities allow us to rewrite trigonometric functions in a simpler form, making them easier to integrate.

4. How do you use substitution in trigonometric integration?

Substitution is a key technique in trigonometric integration. It involves replacing the variable in the integral with a new variable, often chosen to simplify the integral. For example, we may use the substitution u = sin(x) to transform an integral involving sine into one involving only u, which is easier to solve.

5. Are there any tips for solving trigonometric integrals?

There are several tips that can help when solving trigonometric integrals. These include using trigonometric identities to simplify the integral, choosing the right substitution to make the integral easier to solve, and breaking the integral into smaller parts. It is also helpful to practice and familiarize oneself with common integrals involving trigonometric functions.

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