How can I integrate these two challenging expressions?

In summary, the conversation is about solving two integration questions. The first question involves factoring and using a half outside to simplify the integral, while the second question suggests using a substitution method by letting u=3x and considering the derivative of arctan u.
  • #1
Joza
139
0
I need help on these two integration questions:

1.

(x^2 -16)/(2x + 8) dx


2.

1/(1+9x^2) dx


I can't seem to find a "u" for either. And I tried a tan identity on number 2 to no avail.
 
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  • #2
Did you try factoring the first one?
 
  • #3
OH WAIT...

Yea so x^2 - 16 would be (x + 4)(x - 4).

Then I can pull a 1/2 outside, and I'll be left with (x + 4) on the bottom. That will cancel out, so it will be the integral of (x - 4)?
 
  • #4
Well, because of that 1/2 you "pulled outside", it will be (1/2) the integral of x-4.

As far as the second is concerned, you might try letting u= 3x and then think about the derivative of arctan u.
 
  • #5
Yea that's what I meant.

Well I tried using 1/(1^2 + (3x)^2)...but that didnt work.
 

1. What is integration?

Integration is a mathematical process that involves finding the area under a curve. It is often used to calculate the total change or accumulation of a quantity over a given interval.

2. What are the different types of integration?

The two main types of integration are indefinite integration and definite integration. Indefinite integration involves finding an antiderivative of a function, while definite integration involves finding the area under a curve between two given points.

3. Why is integration important?

Integration is important in various fields of science and engineering, such as physics, chemistry, and economics. It allows us to solve problems involving rates of change and accumulation, and provides a way to find exact solutions to difficult mathematical problems.

4. What is the process of integration?

The process of integration involves finding the antiderivative of a function and then evaluating it at the given limits or boundaries. This results in a numerical value that represents the area under the curve.

5. What are some common applications of integration?

Integration is commonly used in physics to calculate displacement, velocity, and acceleration. It is also used in economics to calculate total revenue and total cost, and in statistics to calculate probabilities and expected values.

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