What trick is used to integrate this?

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In summary, the conversation discusses a technique for solving integrals involving e^x, and the question is asked about the origin of the identity \int e^x[f(x) + f'(x)] dx = e^x f(x) + C. The answer suggests using the product rule for integration, which is taught in second-semester single-variable calculus classes.
  • #1
flyingpig
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Homework Statement



[tex]\int e^x \frac{1 + sin(x)}{1 + cos(x)}\;dx[/tex]


The Attempt at a Solution



Apparently there is a trick involving e^x integrals like this

[tex]\int e^x[f(x) + f'(x)] dx = e^x f(x) + C[/tex]

Now my question is not how to compute the integral above but where did this [tex]\int e^x[f(x) + f'(x)] dx = e^x f(x) + C[/tex] identity came from?? How does one know this exists?
 
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  • #2
Take the derivative of the result. That may help you see how this works.
 
  • #3
Yeah, you get this result with integration "by parts", which is the name for using the product rule to rewrite integrals (hopefully in a form you can recognize and solve). If you take a second-semester single-variable calculus class you'll learn about this.
 

1. What is integration?

Integration is a mathematical process of finding the area under a curve. It involves dividing a shape into smaller, known sections and adding them together to find the total area.

2. What is the purpose of integration?

The purpose of integration is to solve problems involving continuous change. It is used in various fields such as physics, engineering, economics, and statistics to analyze and predict behavior.

3. What is the trick to integrate a polynomial function?

The trick to integrating a polynomial function is to use the power rule, which states that the integral of a term with a variable raised to a power is equal to the variable raised to the next highest power, divided by that power.

4. How do you integrate a trigonometric function?

To integrate a trigonometric function, you can use trigonometric identities or substitution. The most commonly used identity is the Pythagorean identity (sin^2x + cos^2x = 1), which can be used to rewrite the function in a more manageable form.

5. Are there any tips for solving integration problems?

Yes, there are several tips for solving integration problems. One tip is to always check for a known function or identity that can be used to simplify the integral. Another tip is to familiarize yourself with common integration techniques such as u-substitution and integration by parts.

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