Chapter 4: Q4P (page 237)

Use L’Hospital’s rule to evaluate $\lim_{x \to 2} \frac{1}{x - 2} dt \int_{2}^{x} \frac{sint}{t} dt$ .

Short Answer

Expert verified

The value of $\lim_{x \to 2} \frac{1}{x - 2} dt \int_{2}^{x} \frac{sint}{t} dt$ is $\frac{1}{2} \sin 2$

Step by step solution

Given Information

Given that $\lim_{x \to 2} \frac{1}{x - 2} dt \int_{2}^{x} \frac{sint}{t} dt$ …... (1)

Finding limx→21x-2dt∫2xsinttdt=6

Use L-Hospital’s rule to differentiate the numerator and denominator separately.

Let $\frac{d}{d x} \int_{2}^{x} \frac{s i n t}{t} d t$

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)

Compare equation (1) and equation (2), then the given function becomes

$\frac{d}{d x} \int_{2}^{x} \frac{\sin t}{t} d t = \frac{\sin x}{x} \cdot \frac{d}{d x} (x) - \frac{\sin 2}{2} \cdot \frac{d}{d x} (2) + \int_{2}^{x} 0 . d t \frac{d}{d x} \int_{2}^{x} \frac{\sin t}{t} d t = \frac{\sin x}{x} (1) - \frac{\sin 2}{2} (0) \frac{d}{d x} \int_{2}^{x} \frac{\sin t}{t} d t = \frac{\sin x}{x}, a n d \frac{d}{d x} (x - 2) = 1$

Then $\frac{1}{x - 2} \int_{2}^{x} \frac{\sin t}{t} d t = \frac{\sin x}{x}$

Therefore,

$\lim_{x \to 2} \frac{1}{x - 2} dt \int_{2}^{x} \frac{sint}{t} dt = \lim_{x \to 2} \frac{\sin x}{x} \lim_{x \to 2} \frac{1}{x - 2} dt \int_{2}^{x} \frac{sint}{t} dt = \frac{1}{2} \sin 2$

Hence, $\lim_{x \to 2} \frac{1}{x - 2} dt \int_{2}^{x} \frac{sint}{t} dt = \frac{1}{2} \sin 2$ .

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Use L’Hospital’s rule to evaluate $\lim_{x \to 2} \frac{1}{x - 2} dt \int_{2}^{x} \frac{sint}{t} dt$ .

Short Answer

Step by step solution

Given Information

Finding limx→21x-2dt∫2xsinttdt=6

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Use L’Hospital’s rule to evaluate limx→21x-2dt∫2xsinttdt.

Short Answer

Step by step solution

Given Information

Finding limx→21x-2dt∫2xsinttdt=6

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Magnetism

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Study anywhere. Anytime. Across all devices.

Use L’Hospital’s rule to evaluate $\lim_{x \to 2} \frac{1}{x - 2} dt \int_{2}^{x} \frac{sint}{t} dt$ .