: Integral of (n+1)th derivative

In summary, the conversation discusses how to solve the equation for the integral of the (n+1)th derivative. The solution involves using the hint to integrate by parts and then using induction to show that the equation holds. The final step is to compute the integral for n=1 and then apply the integration by parts method. The conversation ends with a note of gratitude for the explanation.
  • #1
kfdleb
6
0
URGENT: Integral of (n+1)th derivative

Homework Statement



let f(n+1) be integrable on [a;b]; show that

f(b)=[tex]\sum[/tex] [tex]\frac{f(r)(a)}{r!}[/tex](b-a)r +[tex]\frac{1}{n!}[/tex] [tex]\int^{a}_{b}[/tex]f(n+1)(t)(b-t)ndthint:integrate by parts and use inductionPLEASE any idea about how to solve it would be really appreciated... I've been trying for more than an hour but no idea
 
Last edited:
Physics news on Phys.org
  • #2


Well use the hint. Induct on n. For n = 1, show the equation holds by computing the integral that you get on the RHS after setting n = 1. This is fairly straightforward.
 
  • #3


Start by actually doing the integration by parts. Treat the integral as u*dv where u=f^(n+1)(t) and dv=(b-t)^n*dt. Once you've got that straight then start worrying about the induction.
 
  • #4


10x a lot
 

FAQ: : Integral of (n+1)th derivative

What is the meaning of the integral of (n+1)th derivative?

The integral of (n+1)th derivative is a mathematical concept that represents the area under the curve of the (n+1)th derivative of a function. It is computed by taking the antiderivative of the (n+1)th derivative. This integral is useful in many areas of science, particularly in physics and engineering.

How is the integral of (n+1)th derivative different from a regular integral?

The integral of (n+1)th derivative is different from a regular integral because it involves taking the antiderivative of a higher order derivative. This means that the resulting function will have a higher degree polynomial than the original function, and it may also have multiple constants of integration.

What is the purpose of finding the integral of (n+1)th derivative?

The purpose of finding the integral of (n+1)th derivative is to gain a better understanding of the behavior of a function. It can also help in solving differential equations and in finding the maximum and minimum values of a function.

Can the integral of (n+1)th derivative be negative?

Yes, the integral of (n+1)th derivative can be negative. This can happen when the function has a negative slope and the area under the curve is below the x-axis. In this case, the integral will be negative, indicating that the function is decreasing over that interval.

How can the integral of (n+1)th derivative be used in real-world applications?

The integral of (n+1)th derivative has many real-world applications, such as in physics, engineering, and economics. It can be used to calculate the work done by a force, the velocity of an object, and the change in temperature over time. It is also used in optimization problems to find the maximum or minimum value of a function.

Similar threads

Avoid unpleasant integrals in solving IVP
Replies
3
Views
1K
Calculate this Integral around the Circular Path using Green's Theorem
Replies
3
Views
1K
Prove that this series converges uniformly on R
Replies
4
Views
1K
Evaluate limit of this integral using positive summability kernels
Replies
2
Views
844
Integrating vector-valued functions along curves
Replies
5
Views
1K
Find f(x,y) given partial derivative and initial condition
Replies
6
Views
1K
Given unit impulse response, obtain differential equation
Replies
7
Views
1K
Integral of x^n using Reimann sums
Replies
4
Views
1K
Show uniqueness in Dirichlet problem in unit disk
Replies
6
Views
1K
On pointwise convergence of Fourier series
Replies
3
Views
1K

Hot Threads

Recent Insights

Back
Top