Strategies for Evaluating Trigonometric Integrals

  • Thread starter Pseudo Statistic
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In summary, the conversation is about evaluating integrals and finding closed forms for them. The first integral (x sin x)/(1 + cos^2 x) d x has no closed form, but its numerical value is approximately -2.4674011... The second integral (2t^2 + 3t*t^3) d(t^2) can be solved by replacing d (t^2) with 2 t dt. The conversation also includes a discussion about a particular substitution and the use of "I" in the calculations.
  • #1
Pseudo Statistic
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Hi,
I'm having a bit of trouble evaluating this integral: (This is not a homework problem btw)
Integral between Pi and 0 (x sin x)/(1 + cos^2 x) d x
I don't even know where to begin with the substitutions or anything... I was thinking the denominator could be sin^2 x... only when I realized this is trig and not hyperbolic. :(

Also, something in my book:
Integral (2t^2 + 3t*t^3) d(t^2)
This was after the substitution of dy to d(t^2) when dealing with a line integral; I don't understand... how do you integrate this?

Thanks for any ideas.
 
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  • #2
I don't think you're going to find a closed form for your integral. If it's any help, the numerical value of your integral is approximately -2.4674011...

For the second integral, replace [itex]d (t^2)[/itex] with [itex]2 t dt[/itex].
 
  • #3
Hi,
Thanks for the reply,
My book shows something weird with the integral...
Here's the work... (Assuming everything I have below has the limits pi and 0)
Let x = pi y, then:
integral (xsin x)/(1 + cos^2 x) dx = ---> integ (pi-y)(sin y)/(1 + cos^2 y) dy = pi * integ (sin y)/(1 + cos^2 y) dy - integ (y sin y)/(1 + cos^2 y) dy = -pi * integ d(cos y)/(1 + cos^2 y) - I = -pi*arctan(cos y) - I = Pi^2 /2 - I
And as it says:
i.e. I = Pi^2 / 2 - I or I = Pi^2 / 4
I got lost at the point where I showed the arrow (And I'm confused about why they chose that particular substitution)...
Also, what's up with this - I thing? Can somebody explain it?
Thanks loads for any replies.
 
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  • #4
Oh, I like that! Basically, it's a simple transformation of the integral with the result that the original integral = some other integral - the original integral and, luckily "some other integral" can be evaluated!
 
  • #5
U can part integrate

[tex] \int_{\pi}^{0} x \frac{-d(\cos x)}{1+\cos^{2}x} =...[/tex]

Daniel.
 

What is the purpose of evaluating integrals?

The purpose of evaluating integrals is to find the exact value of the area under a curve, which can be represented by a mathematical function. This is useful in many fields of science and engineering, such as physics, chemistry, and economics, where the integration process can provide important insights and solutions to real-world problems.

What are the different methods for evaluating integrals?

There are several methods for evaluating integrals, including the substitution method, integration by parts, and partial fractions. Each method is useful for different types of integrals and can help simplify the integration process. Additionally, numerical methods such as the trapezoidal rule or Simpson's rule can be used to approximate the value of an integral.

How do I know which method to use for a specific integral?

In most cases, the choice of method for evaluating an integral depends on the form of the integrand. For example, the substitution method is useful for integrals involving trigonometric functions, while integration by parts is effective for integrals involving products of functions. It is important to practice and familiarize yourself with different integration techniques to determine which method is best suited for a particular integral.

What are the common mistakes to avoid when evaluating integrals?

Some common mistakes to avoid when evaluating integrals include forgetting to apply the chain rule or product rule, using incorrect substitution variables, and making errors in algebraic simplification. It is also important to check for any discontinuities or singularities in the integrand that may require special treatment. It is helpful to double-check your work and use multiple methods to verify your solution.

How can I improve my skills in evaluating integrals?

To improve your skills in evaluating integrals, it is important to practice regularly and familiarize yourself with different integration techniques. You can also seek guidance from a mentor or tutor, and use online resources and textbooks for additional practice problems. Additionally, understanding the underlying concept of integration and its applications can help you become more proficient in evaluating integrals.

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