Q4P Use L鈥橦ospital鈥檚 rule to eva... [FREE SOLUTION]

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Chapter 4: Q4P (page 237)

Use L鈥橦ospital鈥檚 rule to evaluate $\lim_{x 鈫� 2} \frac{1}{x - 2} dt {鈭�}_{2}^{x} \frac{sint}{t} dt$ .

Short Answer

Expert verified

The value of $\lim_{x 鈫� 2} \frac{1}{x - 2} dt {鈭�}_{2}^{x} \frac{sint}{t} dt$ is $\frac{1}{2} \sin 2$

Step by step solution

Given Information

Given that $\lim_{x 鈫� 2} \frac{1}{x - 2} dt {鈭�}_{2}^{x} \frac{sint}{t} dt$ 鈥�... (1)

Finding limx→21x-2dt∫2xsinttdt=6

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

Let $\frac{d}{d x} {鈭�}_{2}^{x} \frac{s i n t}{t} d t$

We know that $\frac{d}{d x} {鈭�}_{u (x)}^{v (x)} f (x, t) d t = f (x, v) \frac{d v}{d x} - f (x, u) \frac{d u}{d x} + {鈭�}_{u}^{v} \frac{鈭� f}{鈭� x} d t$ 鈥�... (2)

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

$\frac{d}{d x} {鈭�}_{2}^{x} \frac{\sin t}{t} d t = \frac{\sin x}{x} 路 \frac{d}{d x} (x) - \frac{\sin 2}{2} 路 \frac{d}{d x} (2) + {鈭�}_{2}^{x} 0 . d t \frac{d}{d x} {鈭�}_{2}^{x} \frac{\sin t}{t} d t = \frac{\sin x}{x} (1) - \frac{\sin 2}{2} (0) \frac{d}{d x} {鈭�}_{2}^{x} \frac{\sin t}{t} d t = \frac{\sin x}{x}, a n d \frac{d}{d x} (x - 2) = 1$

Then $\frac{1}{x - 2} {鈭�}_{2}^{x} \frac{\sin t}{t} d t = \frac{\sin x}{x}$

Therefore,

$\lim_{x 鈫� 2} \frac{1}{x - 2} dt {鈭�}_{2}^{x} \frac{sint}{t} dt = \lim_{x 鈫� 2} \frac{\sin x}{x} \lim_{x 鈫� 2} \frac{1}{x - 2} dt {鈭�}_{2}^{x} \frac{sint}{t} dt = \frac{1}{2} \sin 2$

Hence, $\lim_{x 鈫� 2} \frac{1}{x - 2} dt {鈭�}_{2}^{x} \frac{sint}{t} dt = \frac{1}{2} \sin 2$ .

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91影视

Use L鈥橦ospital鈥檚 rule to evaluate $\lim_{x 鈫� 2} \frac{1}{x - 2} dt {鈭�}_{2}^{x} \frac{sint}{t} dt$ .

Short Answer

Step by step solution

Given Information

Finding limx→21x-2dt∫2xsinttdt=6

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91影视

Use L鈥橦ospital鈥檚 rule to evaluate limx鈫�21x-2dt鈭�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

Atoms and Radioactivity

Thermodynamics

Geometrical and Physical Optics

Radiation

Measurements

Conservation of Energy and Momentum

Study anywhere. Anytime. Across all devices.

Use L鈥橦ospital鈥檚 rule to evaluate $\lim_{x 鈫� 2} \frac{1}{x - 2} dt {鈭�}_{2}^{x} \frac{sint}{t} dt$ .