Chapter 4: Q24MP (page 239)

Find $\frac{d}{d x} \int_{t = \frac{1}{x}}^{t = \frac{2}{x}} \frac{c o s h x t}{t} d t$ .

Short Answer

Expert verified

The value of $\frac{d}{d x} \int_{t = \frac{1}{x}}^{t = \frac{2}{x}} \frac{\cosh x t}{t} d t is 0$ .

Step by step solution

Given Information

Given that $\frac{d}{d x} \int_{t = \frac{1}{x}}^{t = \frac{2}{x}} \frac{\cosh x t}{t} d t$ . ...(1)

Formula Used

We know that $\frac{d}{d x} \int_{u (x)}^{v (x)} f (x, t) d t = f (x, v) \frac{d v}{d x} - f (x, u) \frac{d u}{d x} + \int_{u}^{v} \frac{\partial f}{\partial x} d t$ ...(2)

Solving the given equation

Compare equation (1) and equation (2).

$\frac{d}{d x} \int_{t = \frac{1}{x}}^{t = \frac{2}{x}} \frac{\cosh x t}{t} d t = \frac{\cosh x (\frac{2}{x})}{\frac{2}{x}} \times \frac{d}{d x} (\frac{2}{x}) - \frac{\cosh x (\frac{1}{x})}{\frac{1}{x}} \times \frac{d}{d x} (\frac{1}{x}) + \int_{t = \frac{1}{x}}^{t = \frac{2}{x}} \frac{\sinh x t}{t} d t \frac{d}{d x} \int_{t = \frac{1}{x}}^{t = \frac{2}{x}} \frac{\cosh x t}{t} d t = \cosh 2 \times \frac{x}{2} (\frac{- 2}{x^{2}}) - \cosh 1 \times (x) \times (\frac{- 1}{x^{2}}) + {(\frac{\cosh x t}{x})}_{\frac{1}{x}}^{\frac{2}{x}} \frac{d}{d x} \int_{t = \frac{1}{x}}^{t = \frac{2}{x}} \frac{\cosh x t}{t} d t = (\frac{- 1}{x}) \cosh 2 - \cosh 1 (\frac{- 1}{x}) + \frac{1}{x} [\cosh x (\frac{2}{x}) - \cosh x (\frac{1}{x})]$

Solving Further,

$\frac{d}{d x} \int_{t = \frac{1}{x}}^{t = \frac{2}{x}} \frac{\cosh x t}{t} d t = (\frac{- 1}{x}) \cosh 2 - (\frac{- 1}{x}) \cosh 1 + \frac{1}{x} \cosh 2 - \frac{1}{x} \cosh 1 \frac{d}{d x} \int_{t = \frac{1}{x}}^{t = \frac{2}{x}} \frac{\cosh x t}{t} d t = - \frac{1}{x} \cosh 2 + \frac{1}{x} \cosh 1 + \frac{1}{x} \cosh 2 - \frac{1}{x} \cosh 1 \frac{d}{d x} \int_{t = \frac{1}{x}}^{t = \frac{2}{x}} \frac{\cosh x t}{t} d t = 0$

Hence the value of $\frac{d}{d x} \int_{t = \frac{1}{x}}^{t = \frac{2}{x}} \frac{\cosh x t}{t} d t = 0$ .

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Find $\frac{d}{d x} \int_{t = \frac{1}{x}}^{t = \frac{2}{x}} \frac{c o s h x t}{t} d t$ .

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Solving the given equation

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Find $\frac{d}{d x} \int_{t = \frac{1}{x}}^{t = \frac{2}{x}} \frac{c o s h x t}{t} d t$ .