Help with Integral: c*Sin[ArcTan[f(t)]]

  • Thread starter FunkyDwarf
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    Integral
In summary, the conversation is about integrating the function c*sin(arctan(f(t))) and using substitution to solve for sin(arctan(x)). The participant is initially struggling with the problem but eventually realizes how to solve it by applying inverse sine to the top and bottom equally.
  • #1
FunkyDwarf
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Hey guys,

im completely stumped on this one. I had to go through several substitutions just to get it to this stage which is uninspiriring :P

Integral wrt t of c*Sin[ArcTan[(f(t)]] where f(t) is a linear function of t

Sorry, i can't use latex to save my life :p
-G
 
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  • #2
[tex]\int c*\sin{(\arctan{f(t))}} dt[/tex]?
 
  • #3
Remember:
[tex]x=tan(\arctan(x))=\frac{\sin(\arctan(x))}{\sqrt{1-\sin^{2}(\arctan(x)}}[/tex]

Solve for sin(arctan(x))
 
  • #4
Ah yeh of course, i was worried about applying inverse sin to the top and bottom equally but you sort of can because you know what sine is from the substitution and so can get cos. Thanks!
 

1. What is the purpose of the constant c in the integral?

The constant c represents the arbitrary constant of integration. It is added to the solution of the integral because the derivative of a constant is always equal to zero, so it can be added or subtracted without changing the value of the integral.

2. How do I evaluate the integral of c*Sin[ArcTan[f(t)]]?

To evaluate the integral, you can use the substitution method. Let u = ArcTan[f(t)] and du = f'(t)/(1 + [f(t)]^2) dt. After substituting these values into the integral, it can be simplified to c*Sin(u) du. Then, you can use the power reduction formula for Sin(u) to find the final solution.

3. Can I use a calculator to find the solution of the integral?

Yes, most scientific calculators have a built-in integral function that can solve this type of integral. However, it is important to understand the steps and concepts behind the solution rather than solely relying on the calculator.

4. Are there any special cases or restrictions for this type of integral?

Yes, when evaluating the integral, you should consider the domain of the function f(t) and make sure that it is not undefined at any point. Additionally, for the solution to be valid, the integral should converge.

5. Can this type of integral be solved using other methods?

Yes, besides using the substitution method, this type of integral can also be solved using integration by parts or trigonometric identities. However, the most efficient method may vary depending on the specific integral and its components.

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