Two integrals that I don't know how to solve

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In summary, the student is struggling with two integrals and is unsure if integration by parts is the correct approach. The responder advises against using integration by parts and suggests using basic trig identities for the first integral and polynomial long division or splitting the integral into two parts for the second. They also remind the student to not forget the "dx" when writing integrals.
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vsportsguy
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Homework Statement


There are two integrals that I don't know how to solve. I'm fairly certain the preceding work that led me to these integrals is correct.


Homework Equations


1. ∫sin (x) / (csc (x) + cot (x))
2. ∫x^(2) / (1 - x)


The Attempt at a Solution


U substitution didn't work for either integral. Is integration by parts the correct way to go?

Thank You
 
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  • #2


vsportsguy said:

Homework Statement


There are two integrals that I don't know how to solve. I'm fairly certain the preceding work that led me to these integrals is correct.


Homework Equations


1. ∫sin (x) / (csc (x) + cot (x))
2. ∫x^(2) / (1 - x)


The Attempt at a Solution


U substitution didn't work for either integral. Is integration by parts the correct way to go?

Thank You

Integration by parts is NOT the way to go!

Also, don't forget the "dx". As the integrals get more complicated, omitting this symbol will definitely cause problems.

For the first integral, use the basic trig identities to get everything in terms of sine and cosine.

For the second, one approach is to use polynomial long division to divide x2 by 1 - x. Or, you can write x2/(1 - x) as (x2 - 1 + 1)/(1 - x), and split into two integrals.
 

1. How can I determine the limits of integration for a two-integral problem?

The limits of integration for a two-integral problem are typically determined by the boundaries of the region being integrated over. This can be visualized by plotting the region and identifying the x and y coordinates of the boundaries. In some cases, symmetry or other properties of the integrand can also help in determining the limits.

2. What techniques can I use to solve a two-integral problem?

Some common techniques for solving two-integral problems include substitution, integration by parts, and partial fractions. It is also helpful to consider the properties of the integrand, such as symmetry and periodicity, in choosing an appropriate method.

3. How do I know if a two-integral problem can be solved analytically?

Sadly, there is no fool-proof method for determining if a two-integral problem can be solved analytically. However, some indications that a problem may be solvable analytically include simple, well-known integrands and symmetrical or periodic functions.

4. Can numerical methods be used to solve a two-integral problem?

Yes, numerical methods such as the trapezoidal rule, Simpson's rule, and Monte Carlo integration can be used to approximate the solution to a two-integral problem. These methods are particularly useful when the integrand cannot be solved analytically or when the limits of integration are difficult to determine.

5. How can I check if my solution to a two-integral problem is correct?

One way to check the accuracy of your solution is to use a computer algebra system or online calculator to perform the integration and compare the results. You can also try plugging in some sample values for the variables to see if your solution matches the expected output. Additionally, it is always a good idea to double-check your work and make sure you followed all the steps correctly.

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