Trigonometric Integral Problem

In summary: Thanks for catching that! In summary, the homework statement is that the definite integral of function dx/x^2+1 on interval [tan(3), tan(6)] is incorrect. The trigonometric inverse function for dx/x^2+1 is arctanx+c, but it is incorrect because arctan(tan(3))>arctan(tan(6)). I solved the problem by switching to degrees and it worked.
  • #1
Ahlahn
4
0

Homework Statement



Evaluate the definite integral of function dx/x^2+1 on interval [tan(3), tan(6)]

Homework Equations





The Attempt at a Solution



The trigonometric inverse function for dx/x^2+1 is arctanx+c.
I plugged in tan(3) and tan(6), and subtracted arctan(tan(3)) from arctan(tan(6)) and got the answer -.1415. But it's incorrect...

What am I doing wrong?
 
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  • #2
arctan( tan(x)) = x

Use this.

Your calculator may be in radians or something.
 
  • #3
I switched it to degrees and it worked. Wow. Thanks! What's wrong with radians?!
 
  • #4
╔(σ_σ)╝ said:
arctan( tan(x)) = x

Use this.

Your calculator may be in radians or something.

Noo. The answer IS approximately -.1415. What's wrong is that arctan(tan(x))=x. That's only true if x is in (-pi/2,pi/2) which is the range of arctan. Otherwise arctan(tan(x)) will give you the number 'a' in (-pi/2,pi/2) such that tan(x)=tan(a). It will differ from x by a multiple of pi. -.1415 may be coming up wrong because it's not the exact answer. You can find an exact answer. Your 'interval' is also a little funny. tan(3)>tan(6). That's why your answer is negative.
 
  • #5
Ahlahn said:
I switched it to degrees and it worked. Wow. Thanks! What's wrong with radians?!

Ok. I was diagnosing the wrong problem. There is nothing wrong with radians. The only problem is 3 radians isn't equal to 3 degrees. The problem should have specified that 3 meant degrees.
 
  • #6
Dick said:
Noo. The answer IS approximately -.1415. What's wrong is that arctan(tan(x))=x. That's only true if x is in (-pi/2,pi/2) which is the range of arctan. Otherwise arctan(tan(x)) will give you the number 'a' in (-pi/2,pi/2) such that tan(x)=tan(a). It will differ from x by a multiple of pi. -.1415 may be coming up wrong because it's not the exact answer. You can find an exact answer. Your 'interval' is also a little funny. tan(3)>tan(6). That's why your answer is negative.

Mistake of mine :-(. I gave a hurried answer without checking.
 
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FAQ: Trigonometric Integral Problem

1. What is a trigonometric integral problem?

A trigonometric integral problem is a mathematical problem that involves finding the integral of a trigonometric function, such as sine, cosine, or tangent. It typically involves using various trigonometric identities and integration techniques to find the solution.

2. How do I solve a trigonometric integral problem?

To solve a trigonometric integral problem, you need to use integration techniques such as substitution, integration by parts, or trigonometric identities. It is also important to have a good understanding of the properties and rules of trigonometric functions.

3. What is the importance of trigonometric integrals?

Trigonometric integrals are important in many areas of mathematics and science, including physics, engineering, and geometry. They are used to solve many real-world problems involving periodic or oscillatory functions.

4. Are there any tips for solving trigonometric integral problems?

Some tips for solving trigonometric integral problems include: using trigonometric identities to simplify the integral, looking for patterns in the trigonometric function, and practicing different integration techniques.

5. Can trigonometric integral problems be solved using software or calculators?

Yes, there are many software programs and calculators that can solve trigonometric integral problems. However, it is still important to have a good understanding of the concepts and techniques involved in solving these problems.

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