# An identity to prove using calculus 1

• 0kelvin
In summary: Teacher said we could get 1 point for solving it.In summary, the student is having trouble solving a homework problem and is wondering if they remembered to copy something from the blackboard. The student agrees that one of the terms in the function they are trying to solve may be ##f'(x)## and they need to take the derivative of the function to differentiate it. After doing some calculations, they find that the function satisfies the equation ##f(x) + e^{-x^2} f(x) = x.
0kelvin
Homework Statement
There is this function $$f(x) = e^{-A(x)} + e^{-A(x)} \int_0^x t e^{A(t)} dt$$

I have to prove that f satisfies $$f(x) + e^{-x^2} f(x) = x$$
Relevant Equations
$$A(x) = \int_0^x e^{-y^2} dy$$
I have a feeling that I forgot to copy something from the black board, maybe some f' because as it is I'm not seeing a solution.

I agree, I think one of those ##f(x)## terms in the thing you're supposed to prove is probably ##f'(x)##

I would start by taking the derivative of ##f(x)## with respect to ##x##. That means you're going to need the Fundamental Theorem of Calculus in a couple of places.

Ah right. ##f'(x) + e^{-x^2}f(x) = x##.

Doing all the calculatiosn right, I'll end up with ## xe^{A(x) - A(x)} = x ##. Some chain and ´product rule here and there.

0kelvin said:
Homework Statement: There is this function $$f(x) = e^{-A(x)} + e^{-A(x)} \int_0^x t e^{A(t)} dt$$

I have to prove that f satisfies $$f(x) + e^{-x^2} f(x) = x$$
Homework Equations: $$A(x) = \int_0^x e^{-y^2} dy$$

I have a feeling that I forgot to copy something from the black board, maybe some f' because as it is I'm not seeing a solution.
Sorry to nitpick, but , any conditions on A(x)? Continuity, differentiability, etc? Most likely even nicer, but just for the sake of completeness.

Teacher didn't specify. For some reason he thought it would be a good idea to give everyone 15 minutes to solve it last week. Solving it would give up to +1 point in the next exam.

0kelvin said:
Teacher didn't specify. For some reason he thought it would be a good idea to give everyone 15 minutes to solve it last week. Solving it would give up to +1 point in the next exam.
But in order to differentiate, you would need to know something about A(t). I guess it is assumed that it is nice-enough to be differentiable.

WWGD said:
But in order to differentiate, you would need to know something about A(t). I guess it is assumed that it is nice-enough to be differentiable.
It is given that ##A(x)=\int_0^x e^{-y^2}dy## it is listed under homework equations tab in OP.

WWGD and PeroK
Delta2 said:
It is given that ##A(x)=\int_0^x e^{-y^2}dy## it is listed under homework equations tab in OP.
My bad, did not pay attention.

## What is an identity to prove using calculus 1?

An identity is a mathematical statement that is true for all values of the variables involved. In calculus, identities are often used to simplify or solve equations and can involve various operations such as differentiation, integration, and limits.

## Why is it important to prove identities using calculus 1?

Proving identities in calculus helps to deepen our understanding of mathematical concepts and principles. It also allows us to develop problem-solving skills and apply calculus techniques to more complex equations and problems.

## What are some common identities used in calculus 1?

Some common identities used in calculus 1 include the power rule, product rule, quotient rule, chain rule, and integration by parts. These identities involve differentiating and integrating various types of functions.

## How can calculus 1 be used to prove an identity?

In order to prove an identity using calculus 1, we need to manipulate the given equation using algebraic and calculus techniques, such as substitution, differentiation, or integration. By simplifying both sides of the equation and showing that they are equivalent, we can prove the identity.

## What are some tips for successfully proving an identity using calculus 1?

Some tips for successfully proving an identity using calculus 1 include understanding the basic rules and properties of calculus, breaking down the problem into smaller steps, and practicing with different types of identities. It is also helpful to double-check your work and make sure that both sides of the equation are equivalent before concluding that the identity has been proven.

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