Q30MP Evaluate d2dx2鈭�0x鈭�0xfs,tdsdt... [FREE SOLUTION]

Chapter 4: Q30MP (page 240)

Evaluate $\frac{d^{2}}{{dx}^{2}} {鈭�}_{0}^{x} {鈭�}_{0}^{x} f (s, t) dsdt$ .

Short Answer

Expert verified

Thus, . $\frac{d^{2}}{d x^{2}} {鈭�}_{0}^{x} {鈭�}_{0}^{x} f (s, t) d s d t = 2 f (x) + {鈭�}_{0}^{x} \frac{鈭� f}{鈭� x} d t + {鈭�}_{0}^{x} \frac{鈭� f}{鈭� x} d s$

Step by step solution

Determine the integral value.

$\frac{d^{2}}{d x^{2}} {鈭�}_{0}^{x} {鈭�}_{0}^{x} f (s, t) d s d t = \frac{d}{d x} [\frac{d}{d x} {鈭�}_{0}^{x} d s {鈭�}_{0}^{x} f (s, t) d t] = \frac{d}{d x} {鈭�}_{0}^{x} f (x, t) d t + {鈭�}_{0}^{x} d s \frac{鈭�}{鈭� x} {鈭�}_{0}^{x} f (s, t) d t = \frac{d}{d x} [{鈭�}_{0}^{x} f (x, t) d t + {鈭�}_{0}^{x} [f (s, x) + {鈭�}_{0}^{x} \frac{鈭� f (s, t)}{鈭� x} d t]]$

So, we get,

$\frac{d^{2}}{d x^{2}} {鈭�}_{0}^{x} {鈭�}_{0}^{x} f (s, t) d s d t = \frac{d}{d x} [{鈭�}_{0}^{x} f (x, t) d t + {鈭�}_{0}^{x} f (s, x) d s] = \frac{d}{d x} [{鈭�}_{0}^{x} f (x, t) d t] + \frac{d}{d x} [{鈭�}_{0}^{x} f (s, x) d s]$

Determine the Solution using formulas.

Now we use the formulas are,

$\frac{d}{d x} {鈭�}_{0}^{x} f (x, t) d t = f (x) + {鈭�}_{0}^{x} \frac{鈭� f}{鈭� x} d t \frac{d}{d x} {鈭�}_{0}^{x} f (s, x) d s = f (x) + {鈭�}_{0}^{x} \frac{鈭� f}{鈭� x} d s$

Substituting these formulas,

$\frac{d^{2}}{d x^{2}} {鈭�}_{0}^{x} {鈭�}_{0}^{x} f (s, t) d s d t = f (x) + {鈭�}_{0}^{x} \frac{鈭� f}{鈭� x} d t + f (x) + {鈭�}_{0}^{x} \frac{鈭� f}{鈭� x} d s = 2 f (x) + {鈭�}_{0}^{x} \frac{鈭� f}{鈭� x} d t + {鈭�}_{0}^{x} \frac{鈭� f}{鈭� x} d s$

So, the solution is,

$\frac{d^{2}}{d x^{2}} {鈭�}_{0}^{x} {鈭�}_{0}^{x} f (s, t) d s d t = 2 f (x) + {鈭�}_{0}^{x} \frac{鈭� f}{鈭� x} d t + {鈭�}_{0}^{x} \frac{鈭� f}{鈭� x} d s$

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91影视

Evaluate $\frac{d^{2}}{{dx}^{2}} {鈭�}_{0}^{x} {鈭�}_{0}^{x} f (s, t) dsdt$ .

Short Answer

Step by step solution

Determine the integral value.

Determine the Solution using formulas.

One App. One Place for Learning.

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91影视

Evaluate d2dx2鈭�0x鈭�0xfs,tdsdt .

Short Answer

Step by step solution

Determine the integral value.

Determine the Solution using formulas.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Oscillations

Mechanics and Materials

Space Physics

Solid State Physics

Atoms and Radioactivity

Magnetism

Study anywhere. Anytime. Across all devices.

Evaluate $\frac{d^{2}}{{dx}^{2}} {鈭�}_{0}^{x} {鈭�}_{0}^{x} f (s, t) dsdt$ .