Animated Solution for Mathematics - Definite Integration: If $f(x)$ satisfies the relation $f(x)=e^x+{\int }_0^1(y+x{e}^x)f(y)dy$, then $e+f(0)$ is equal to .........

The given integral equation is:

Our objective is to determine the value of $e+f(0)$.

When observing the integral ${\int }_0^1(y+x{e}^x)f(y)dy$, note that the variable of integration is $y$. Since the limits are $0$ and $1$, the entire integral evaluates to a constant.

Within this integral, $x$ and $e^x$ are independent of $y$. We can treat them as constants relative to the integration process, which is our first major breakthrough.

Using the linearity of the integral, we expand the expression:

Since $x{e}^x$ is constant with respect to $y$, we pull it out:

Let us define the constants $A$ and $B$ as follows:

Substituting these into our equation, we obtain a manageable algebraic form:

To solve for $A$ and $B$, we substitute $f(y)=(1+Ay)e^y+B$ back into the definition of $A$:

Expanding this integral yields:

Evaluating these standard integrals: 1. ${\int }_0^1{e}^{y}dy=[e^y{]}_0^1=e-1$ 2. ${\int }_0^{1}Bdy=[By{]}_0^1=B$ 3. ${\int }_0^{1}y{e}^{y}dy=[y{e}^y-e^y{]}_0^1=(e-e)-(0-1)=1$

Substituting these values back into the equation for $A$:

The $A$ terms cancel out, simplifying the expression to:

We now find $f(0)$ using our derived formula $f(x)=(1+Ax)e^x+B$:

Substituting $B=1-e$:

Finally, we calculate the requested value $e+f(0)$:

The final answer is 2.