Dealing with functions of several variables

In summary, the given problem asks to find the integral of \frac{t^x - t^y}{\ln t} from 0 to 1, assuming x > -1 and y > -1. The student is unsure of how to approach the problem, but later finds out that the question belongs to a omitted chapter in their class.
  • #1
Nick89
555
0

Homework Statement


Assuming [itex]x > -1[/itex] and [itex]y > -1[/itex], find the following integral:
[tex]\displaystyle \int_0^1 \frac{t^x - t^y}{\ln t} dt[/tex]


Homework Equations





The Attempt at a Solution


I have no idea where to start on this one... It came up in an exam for a class dealing with functions of several variables (eg, f(x,y,z)) so I don't think it's a usual substitution problem...

I tried writing it as [tex]\int t^x \ln(-t) dt[/tex] (minus the same for y) but I couldn't get any further...

Do I perhaps have to find some taylor polynomial or something?
 
Physics news on Phys.org
  • #2


Nick89 said:
I tried writing it as [tex]\int t^x \ln(-t) dt[/tex] (minus the same for y) but I couldn't get any further...
This is wrong that's not the same integral ( do you know why?)

Use IBP.
 
  • #3


Sorry I screwed up there, heh.

Anyway, I just found out that this question in the exam was part of some chapter that is now omitted, that would explain why I have no idea how to start on this one ^^
 

1. What is the definition of a function of several variables?

A function of several variables is a mathematical relationship that maps a set of input values from multiple independent variables to a set of output values. It can be represented by an equation or a graph.

2. How do you determine the domain and range of a function of several variables?

The domain of a function of several variables is the set of all possible input values for the independent variables. The range is the set of all possible output values. To determine the domain and range, you can analyze the equation or graph of the function and identify any restrictions or patterns in the input and output values.

3. What is the concept of partial derivatives in dealing with functions of several variables?

Partial derivatives are the derivatives of a function with respect to one of its variables while holding the other variables constant. They measure the rate of change of the function in a specific direction and are useful in optimization and gradient descent algorithms in multivariable calculus.

4. How do you find critical points of a function of several variables?

The critical points of a function of several variables are the points where the partial derivatives are equal to zero or do not exist. To find them, you can take the partial derivatives of the function and set them equal to zero, then solve the resulting system of equations. Alternatively, you can use the gradient vector to find points where the function has a local maximum or minimum.

5. What is the practical application of functions of several variables in science?

Functions of several variables are widely used in various fields of science, such as physics, engineering, economics, and biology. They are used to model complex systems and relationships between multiple variables, analyze data, and make predictions. Some examples include thermodynamic equations, economic supply and demand functions, and population growth models.

Similar threads

Determine limits of integration in double integral change of variables
    • Calculus and Beyond Homework Help
Replies
2
Views
148
Solving a separable matrix ODE.
    • Calculus and Beyond Homework Help
Replies
3
Views
323
Evaluate the line integral
    • Calculus and Beyond Homework Help
Replies
12
Views
983
Avoid unpleasant integrals in solving IVP
    • Calculus and Beyond Homework Help
Replies
3
Views
566
Change of variables in multiple integrals
    • Calculus and Beyond Homework Help
Replies
34
Views
2K
How to get general solution via Green's function?
    • Calculus and Beyond Homework Help
Replies
1
Views
701
Steady state heat equation in a rectangle with a punkt heat source
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
Can't Find a Correct Method to Integrate \int (t - 2)^2\sqrt{t}\,dt?
    • Calculus and Beyond Homework Help
Replies
9
Views
723
Solving a Partial Differential Equation with the Characteristic Method
    • Calculus and Beyond Homework Help
Replies
5
Views
1K
Solve the given equation using the Lambert W function
    • Calculus and Beyond Homework Help
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top