Question

Evaluate $\frac{{\mathbf{d}}^{\mathbf{2}}}{{\mathbf{dx}}^{\mathbf{2}}}\underset{\mathbf{0}}{\overset{\mathbf{x}}{\int }}\underset{\mathbf{0}}{\overset{\mathbf{x}}{\int }}\mathbf{f}\left(\mathbf{s}\mathbf{,}\mathbf{t}\right)\mathbf{dsdt}$ .

Short answer

Thus, .$\frac{d^2}{d{x}^2}\underset{0}{\overset{x}{\int }}\underset{0}{\overset{x}{\int }}f\left(s,t\right)dsdt=2f\left(x\right)+\underset{0}{\overset{x}{\int }}\frac{\partial f}{\partial x}dt+\underset{0}{\overset{x}{\int }}\frac{\partial f}{\partial x}ds$

Step-by-step solution

Determine the integral value.

$$\begin{gathered} \frac{d^2}{d{x}^2}\underset{0}{\overset{x}{\int }}\underset{0}{\overset{x}{\int }}f\left(s,t\right)dsdt=\frac{d}{dx}\left[\frac{d}{dx}\underset{0}{\overset{x}{\int }}ds\underset{0}{\overset{x}{\int }}f\left(s,t\right)dt\right] \\ =\frac{d}{dx}\underset{0}{\overset{x}{\int }}f\left(x,t\right)dt+\underset{0}{\overset{x}{\int }}ds\frac{\partial }{\partial x}\underset{0}{\overset{x}{\int }}f\left(s,t\right)dt \\ =\frac{d}{dx}\left[\underset{0}{\overset{x}{\int }}f\left(x,t\right)dt+\underset{0}{\overset{x}{\int }}\left[f\left(s,x\right)+\underset{0}{\overset{x}{\int }}\frac{\partial f\left(s,t\right)}{\partial x}dt\right]\right] \end{gathered}$$

So, we get,

$$\begin{gathered} \frac{d^2}{d{x}^2}\underset{0}{\overset{x}{\int }}\underset{0}{\overset{x}{\int }}f\left(s,t\right)dsdt=\frac{d}{dx}\left[\underset{0}{\overset{x}{\int }}f\left(x,t\right)dt+\underset{0}{\overset{x}{\int }}f\left(s,x\right)ds\right] \\ =\frac{d}{dx}\left[\underset{0}{\overset{x}{\int }}f\left(x,t\right)dt\right]+\frac{d}{dx}\left[\underset{0}{\overset{x}{\int }}f\left(s,x\right)ds\right] \end{gathered}$$

Determine the Solution using formulas.

Now we use the formulas are,

$$\begin{gathered} \frac{d}{dx}\underset{0}{\overset{x}{\int }}f\left(x,t\right)dt=f\left(x\right)+\underset{0}{\overset{x}{\int }}\frac{\partial f}{\partial x}dt \\ \frac{d}{dx}\underset{0}{\overset{x}{\int }}f\left(s,x\right)ds=f\left(x\right)+\underset{0}{\overset{x}{\int }}\frac{\partial f}{\partial x}ds \end{gathered}$$

Substituting these formulas,

$$\begin{gathered} \frac{d^2}{d{x}^2}\underset{0}{\overset{x}{\int }}\underset{0}{\overset{x}{\int }}f\left(s,t\right)dsdt=f\left(x\right)+\underset{0}{\overset{x}{\int }}\frac{\partial f}{\partial x}dt+f\left(x\right)+\underset{0}{\overset{x}{\int }}\frac{\partial f}{\partial x}ds \\ =2f\left(x\right)+\underset{0}{\overset{x}{\int }}\frac{\partial f}{\partial x}dt+\underset{0}{\overset{x}{\int }}\frac{\partial f}{\partial x}ds \end{gathered}$$

So, the solution is,

$\frac{d^2}{d{x}^2}\underset{0}{\overset{x}{\int }}\underset{0}{\overset{x}{\int }}f\left(s,t\right)dsdt=2f\left(x\right)+\underset{0}{\overset{x}{\int }}\frac{\partial f}{\partial x}dt+\underset{0}{\overset{x}{\int }}\frac{\partial f}{\partial x}ds$