How to Integrate x*ln(2x) and x^2/(x^2-4)?

  • Thread starter lastdayx52
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    Integrals
In summary: Got a:\frac{x^2*(2*ln(2)-1)}{4}+\frac{x^2*ln(x)}{2} + CChecked on calculator, and was correct. Thank you!
  • #1
lastdayx52
6
0
a. [itex]
\int x*ln(2x) dx
[/itex]
b. [itex]
\int \frac{x^2}{x^2-4} dx
[/itex]

My Attempt:
a. I have abs. no clue how to start it off. Do you start it off using integration by parts?
b. Same with this. I don't know how to start it off.

"Random" Questions:
1) What does "converge" and "diverge" mean? i.e. Another question asks "Determine if each integral converges or diverges". I can do the integrals, but I'm not sure what its asking. One results in the answer being pi/3, the other results in infinity.
2) When the question tells you to setup an integral for a volume of a solid generated when the region is rotated about the line x=-2 using cylindrical shells, you just do:
[itex]\pi *
\int (f(x)+2)^2 dx
[/itex]
Correct?

Thanks!

P.S. I'm not asking for answers, but rather how to start them off. Maybe 1 or 2 beginning steps in integrating these? =D
 
Last edited:
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  • #2
Hi lastdayx52, welcome to PF:smile:

Let's start with a) and b)... For a) try integration by parts...For b), hint: [itex]x^2=(x^2-4)+4[/itex]:wink:

Give it a shot and post your attempt if you require further assistance.
 
  • #3
gabbagabbahey said:
Hi lastdayx52, welcome to PF:smile:

Let's start with a) and b)... For a) try integration by parts...For b), hint: [itex]x^2=(x^2-4)+4[/itex]:wink:

Give it a shot and post your attempt if you require further assistance.

Got a:
[itex]\frac{x^2*(2*ln(2)-1)}{4}+\frac{x^2*ln(x)}{2} + C[/itex]
Checked on calculator, and was correct. Thank you!

Now for b, that doesn't help me at all... >.>
 
  • #4
lastdayx52 said:
Now for b, that doesn't help me at all... >.>

Why not?

[tex]\int \frac{x^2}{x^2-4} dx=\int \frac{(x^2-4)+4}{x^2-4} dx=\int \frac{x^2-4}{x^2-4} dx +\int \frac{4}{x^2-4} dx[/tex]
 
  • #5
gabbagabbahey said:
Hi lastdayx52, welcome to PF:smile:

Let's start with a) and b)... For a) try integration by parts...For b), hint: [itex]x^2=(x^2-4)+4[/itex]:wink:

Give it a shot and post your attempt if you require further assistance.

gabbagabbahey said:
Why not?

[tex]\int \frac{x^2}{x^2-4} dx=\int \frac{(x^2-4)+4}{x^2-4} dx=\int \frac{x^2-4}{x^2-4} dx +\int \frac{4}{x^2-4} dx[/tex]

LOL haha... See I did that, but stupidly thought the integral of 1 = 1, and when checked against the calculator, it was obviously not correct. haha... Thanks! =P
 

Related to How to Integrate x*ln(2x) and x^2/(x^2-4)?

1. How do I determine which method to use when solving integrals?

There are several methods for solving integrals, such as substitution, integration by parts, trigonometric substitution, and partial fractions. The best way to determine which method to use is to look for patterns in the integrand and to practice solving different types of integrals.

2. What is the process for solving integrals?

The process for solving integrals involves using mathematical techniques to find the antiderivative of a function. This can be done by using basic integration rules and methods, such as u-substitution or integration by parts. Once the antiderivative is found, the integral can be evaluated at specific limits to find the definite integral.

3. How do I know if my answer to an integral is correct?

To check if your answer to an integral is correct, you can use the fundamental theorem of calculus which states that the derivative of the antiderivative of a function is equal to the original function. You can also use online integration calculators or check your answer with a trusted source, such as a textbook or a teacher.

4. Are there any special rules for solving trigonometric integrals?

Yes, there are specific rules for solving trigonometric integrals. These include using trigonometric identities, substitution with trigonometric functions, and using integration by parts. It's important to be familiar with these rules when solving trigonometric integrals.

5. What are the common mistakes to avoid when solving integrals?

Some common mistakes to avoid when solving integrals include forgetting to add the constant of integration, using the wrong substitution, and making algebraic errors. It's also important to check for any potential discontinuities or points where the integral may not exist. Additionally, it's important to practice and understand the different rules and methods for solving integrals to avoid making mistakes.

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