Evaluate the definite integral

In summary, the conversation involves a question about integrating the expression \int^{\frac{\pi}{4}}_{0}\frac{1+cos^{2}(x)}{cos^{2}(x)}. The person attempted to solve it using u-substitution, splitting it into two fractions and a trig identity, but was unsure how to integrate the fraction 1/cos^2(x). Another person suggested checking a standard integral for \sec^2(x).
  • #1
crybllrd
120
0

Homework Statement




Homework Equations


The Attempt at a Solution



I tried u-sub, splitting it up into 2 fractions, and a trig identity (cos^2x=(1+cos2x)/2).
Am i missing something? The other problems were quite easy.
 
Physics news on Phys.org
  • #2
What did you get if you split it up in two fractions? Why didn't it work?
 
  • #3
I got two fractions, 1/cos^2(x) and cos^2(x)/cos^2(x)=1

I wasn't sure how to integrate the 1/cos^2(x)
 
  • #5
Wow, see what happens when I take a summer off?
Thanks a lot.
 

FAQ: Evaluate the definite integral

What is the purpose of evaluating a definite integral?

The purpose of evaluating a definite integral is to find the exact numerical value of the area under a curve between two specific points on the x-axis. It is also used to solve problems involving accumulations, such as finding the total distance traveled or total amount of a substance produced.

How is a definite integral different from an indefinite integral?

A definite integral has specific limits of integration, while an indefinite integral does not. This means that a definite integral will give a numerical value, while an indefinite integral will result in a function with a constant of integration.

What are the steps for evaluating a definite integral?

The steps for evaluating a definite integral are:

  1. Identify the limits of integration
  2. Find the antiderivative of the integrand
  3. Substitute the limits of integration into the antiderivative
  4. Find the difference between the two resulting values

Can a definite integral have a negative value?

Yes, a definite integral can have a negative value. This occurs when the area under the curve is below the x-axis, resulting in a negative value for the area. In this case, the definite integral represents the signed area under the curve.

What is the significance of the definite integral in real-world applications?

The definite integral has many real-world applications, such as calculating the work done by a force, finding the average value of a function, and determining the amount of change in a system over a period of time. It is also used in physics, engineering, economics, and other fields to solve problems involving rates of change and accumulations.

Similar threads

Replies
15
Views
1K
Replies
22
Views
2K
Replies
1
Views
1K
Replies
9
Views
869
Replies
9
Views
927
Replies
4
Views
1K
Back
Top