Integration of Inverse Tangent Function

In summary, the problem involves integrating the inverse tangent function with a given expression. The solution involves manipulating the expression to be in the form of \int\frac{1}{a^{2}+x^{2}}dx. By using the substitution (3e^x)^2 for 9e^(2x), the integration can be solved.
  • #1
Lanza52
63
0
[SOLVED] Integration of Inverse Tangent Function

[tex]\int \frac{e^{x}}{4+9e^{2x}}dx[/tex]

Saw the problem, looked at it for a bit. Noticed that it is a inverse Tangent function. Played with some integration by parts and substitution and couldn't figure it out. Can anybody toss me a starting point on this problem?

Thanks
 
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  • #2
Think of 9e^(2x) as (3e^x)^2, that should get you started.
 
  • #3
you're aim should be to put your integral in this form [tex]\int\frac{1}{a^{2}+x^{2}}dx[/tex] so algebraically manipulate it as hotcommodity suggested.
 
  • #4
Ahh...thank you guys =P

For some reason I wasn't letting myself put the 9e^2x as (3e^x)^2. Basic order of operations tricks me again.
 

1. What is the inverse tangent function?

The inverse tangent function, also known as arctangent, is the inverse operation of the tangent function. It is represented by tan-1(x) or arctan(x) and gives the angle whose tangent is equal to the given value.

2. How is the inverse tangent function integrated?

The integral of the inverse tangent function is: ∫ tan-1(x) dx = x tan-1(x) - 1/2 ln(1 + x2) + C, where C is the constant of integration.

3. What is the domain and range of the inverse tangent function?

The domain of the inverse tangent function is all real numbers, while the range is from -π/2 to π/2, or approximately -1.57 to 1.57.

4. What are some applications of the inverse tangent function?

The inverse tangent function is commonly used in trigonometry and geometry to find angles in right triangles. It is also used in physics and engineering to solve problems involving circular motion and oscillations.

5. How is the inverse tangent function related to other trigonometric functions?

The inverse tangent function is the inverse of the tangent function, meaning that tan-1(x) = y if and only if tan(y) = x. It is also related to the other inverse trigonometric functions, such as inverse sine and inverse cosine, through various trigonometric identities and equations.

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