If x(f x)+∫0^x f(z) d z=x e^x, f(1)=e, Then f(2) is (a) (3 e)/(4) (b) (3 e^2)/(4) (c) (4 e^2)/(3

step 1 In this question we have to find the value of function at point 2 and we have given that x f x x

If x(f x)+∫0^x f(z) d z=x e^x, f(1)=e, Then f(2) is (a) (3...

Key Concepts

Fundamental Theorem of Calculus

This theorem connects differentiation with integration. It allows one to differentiate an integral with a variable upper limit, turning the integration operation into an evaluation of the integrand at that upper limit. This is central when dealing with equations where an integral expression depends on the variable of differentiation.

Differentiation of Equations Involving Integrals

When an equation includes an integral with a variable limit, it can be differentiated with respect to that variable to simplify the problem or to obtain a differential equation. This technique leverages the fundamental theorem of calculus to convert the integral term into a function of the variable, making it easier to isolate unknown functions.

Product Rule in Differentiation

The product rule is used to differentiate products of two functions. In equations that involve terms like a function multiplied by the variable, the product rule helps in correctly computing the derivative, ensuring that all parts of the product are accounted for.

First Order Linear Differential Equations

A first order linear differential equation has the form dy/dx + P(x)y = Q(x). Such equations are solved using systematic approaches like finding integrating factors. Recognizing and restructuring an equation into this form is key to finding the solution of the unknown function.

Integrating Factor Method

This technique is specifically used for solving first order linear differential equations. It involves multiplying the entire differential equation by a carefully chosen function (the integrating factor), which transforms the left-hand side of the equation into the derivative of a product. This method simplifies integration and facilitates finding the general solution.

Initial Value Problems

In many differential equations, an initial condition is provided to allow for the determination of an arbitrary constant in the general solution. Initial conditions ensure the solution is unique and applicable to the specific context of the problem by matching the function’s value at a particular point.

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