Chapter 4: Q26MP (page 239)

Find $\frac{d}{d x} \int_{0}^{x^{2}} \frac{s i n x t}{t} d t$ .

Short Answer

Expert verified

The value of $\frac{d}{d x} \int_{0}^{x^{2}} \frac{\sin x t}{t} d t is \frac{3}{x} \sin x^{3}$ .

Step by step solution

Given Information

Given that . $\frac{d}{d x} \int_{0}^{x^{2}} \frac{\sin x t}{t} d t$ ...(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).

$\frac{d}{d x} \int_{0}^{x^{2}} \frac{\sin x t}{t} d t = \frac{\sin (x . x^{2})}{x^{2}} . \frac{d}{d x} (x^{2}) - \frac{\sin (x . 0)}{0} . \frac{d}{d x} (0) + \int_{0}^{x^{2}} t \frac{\cos x t}{t} d t \frac{d}{d x} \int_{0}^{x^{2}} \frac{\sin x t}{t} d t = \frac{\sin x^{3}}{x^{2}} . (2 x) - \frac{\sin 0}{0} . (0) + \int_{0}^{x^{2}} \cos x t d t \frac{d}{d x} \int_{0}^{x^{2}} \frac{\sin x t}{t} d t = 2 x \frac{\sin x^{3}}{x^{2}} + {(\frac{\sin x t}{x})}_{0}^{x^{2}}$

Therefore,

$\frac{d}{d x} \int_{0}^{x^{2}} \frac{\sin x t}{t} d t = 2 x \frac{\sin x^{3}}{x^{2}} + \frac{\sin (x . x^{2})}{x} - \frac{\sin (x . 0)}{x} \frac{d}{d x} \int_{0}^{x^{2}} \frac{\sin x t}{t} d t = \frac{2 \sin x^{3}}{x} + \frac{\sin x^{3}}{x} \frac{d}{d x} \int_{0}^{x^{2}} \frac{\sin x t}{t} d t = \frac{2}{x} \sin x^{3} + \frac{1}{x} \sin x^{3} \frac{d}{d x} \int_{0}^{x^{2}} \frac{\sin x t}{t} d t = \frac{3}{x} \sin x^{3}$

Hence the value of $\frac{d}{d x} \int_{0}^{x^{2}} \frac{\sin x t}{t} d t = \frac{3}{x} \sin x^{3}$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Find $\frac{d}{d x} \int_{0}^{x^{2}} \frac{s i n x t}{t} d t$ .

Short Answer

Step by step solution

Given Information

Formula Used

Solving the given equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electricity

Oscillations

Energy Physics

Quantum Physics

Kinematics Physics

Geometrical and Physical Optics

Study anywhere. Anytime. Across all devices.

Find ddx∫0x2sinxttdt.

Short Answer

Step by step solution

Given Information

Formula Used

Solving the given equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electricity

Oscillations

Energy Physics

Quantum Physics

Kinematics Physics

Geometrical and Physical Optics

Study anywhere. Anytime. Across all devices.

Find $\frac{d}{d x} \int_{0}^{x^{2}} \frac{s i n x t}{t} d t$ .