Chapter 4: Q30MP (page 240)

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

Short Answer

Expert verified

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

Step by step solution

Determine the integral value.

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

So, we get,

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

Determine the Solution using formulas.

Now we use the formulas are,

$\frac{d}{d x} \int_{0}^{x} f (x, t) d t = f (x) + \int_{0}^{x} \frac{\partial f}{\partial x} d t \frac{d}{d x} \int_{0}^{x} f (s, x) d s = f (x) + \int_{0}^{x} \frac{\partial f}{\partial x} d s$

Substituting these formulas,

$\frac{d^{2}}{d x^{2}} \int_{0}^{x} \int_{0}^{x} f (s, t) d s d t = f (x) + \int_{0}^{x} \frac{\partial f}{\partial x} d t + f (x) + \int_{0}^{x} \frac{\partial f}{\partial x} d s = 2 f (x) + \int_{0}^{x} \frac{\partial f}{\partial x} d t + \int_{0}^{x} \frac{\partial f}{\partial x} d s$

So, the solution is,

$\frac{d^{2}}{d x^{2}} \int_{0}^{x} \int_{0}^{x} f (s, t) d s d t = 2 f (x) + \int_{0}^{x} \frac{\partial f}{\partial x} d t + \int_{0}^{x} \frac{\partial f}{\partial x} d s$

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

Short Answer

Step by step solution

Determine the integral value.

Determine the Solution using formulas.

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Evaluate d2dx2∫0x∫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

Turning Points in Physics

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