Solving an Integral without Symmetry: A Challenge

In summary, when approaching solving an integral without symmetry, one can use strategies such as substitution or integration by parts. Common challenges include identifying the appropriate method and dealing with complex functions. An example of solving an integral without symmetry is using the substitution u = x^2 for the integral ∫e^(x^2)dx. Tips for solving integrals without symmetry include looking for patterns and familiar functions, and practicing different techniques. To ensure correctness, one can check their answer by taking the derivative or using online tools.
  • #1
SeReNiTy
170
0
I'm trying to solve a special integral.

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

So far I've tried constructing a function F(a) and differentitating to show that it is a 0, hence any a would result in the same answer. Thus chose a = 2, and solve it quite easily. Although, all my arguments rely on symmetry, just wondering if it is possible to solve without symmetry.
 
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  • #2
don't worry guys, I've worked it out
 

1. How do you approach solving an integral without symmetry?

There are a few strategies you can use to solve an integral without symmetry. One approach is to use the substitution method where you replace the variable in the integral with a new variable. Another approach is to use integration by parts, which involves breaking down the integral into two parts and using the product rule of derivatives.

2. What are some common challenges in solving integrals without symmetry?

Some common challenges include identifying the appropriate substitution or integration by parts, dealing with complex functions, and determining the limits of integration for the new variable.

3. Can you provide an example of solving an integral without symmetry?

Sure, let's say we have the integral ∫e^(x^2)dx. Using the substitution u = x^2, we can rewrite this as ∫e^u du. Then, using the power rule of integration, the integral becomes 1/2 * e^u + C. Substituting back in for u, we get the final answer of 1/2 * e^(x^2) + C.

4. Are there any tips or tricks for solving integrals without symmetry?

One tip is to always start by looking for any patterns or familiar functions within the integral. This can help guide you towards the appropriate substitution or integration by parts. It's also helpful to practice and familiarize yourself with different techniques for solving integrals without symmetry.

5. How do you know if you have solved an integral without symmetry correctly?

You can check your answer by taking the derivative of the integral. If you get back the original integrand, then you have solved it correctly. You can also use online tools or calculators to verify your answer.

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