Q. 46 Use the Second Fundamental Theor... [FREE SOLUTION]

Chapter 4: Q. 46 (page 400)

Use the Second Fundamental Theorem of Calculus, if needed, to calculate each of the derivatives given below.
$\frac{d}{d x} (x^{2} {鈭�}_{0}^{\sin 鈦� x} 鈥� \sqrt{t} d t)$

Short Answer

Expert verified

Ans: $\frac{d}{d x} (x^{2} {鈭�}_{0}^{\sin 鈦� x} 鈥� \sqrt{t} d t) = 2 x \sqrt{\sin 鈦� x} \cos 鈦� x + x^{2} \frac{\cos^{2} 鈦� x}{2 \sqrt{\sin 鈦� x}} 鈭� x^{2} \sin 鈦� x \sqrt{\sin 鈦� x}$

Step by step solution

Step 1. Given information.

given expression,

$\frac{d}{d x} (x^{2} {鈭�}_{0}^{\sin 鈦� x} 鈥� \sqrt{t} d t)$

Step 2. The objective is to calculate the derivative.

Now, if $f$ is continuous on role="math" localid="1648716835455" $[a, b]$ then for allrole="math" localid="1648716841853" $x 鈭� [a, b]$ ,

$\frac{d}{d x} {鈭�}_{a}^{u (x)} 鈥� f (t) d t = f (u (x)) u^{鈥�} (x)$

So,

$\begin{aligned} f (u (t)) = \sqrt{\sin 鈦� t} \\ f (u (x)) = \sqrt{\sin 鈦� x} \\ u^{鈥�} (x) = \cos 鈦� x \\ f (u (x)) u^{鈥�} (x) = \sqrt{\sin 鈦� x} \cos 鈦� x \end{aligned}$

Step 3. The derivate expression can be written as,

$\begin{aligned} \frac{d}{d x} (x^{2} {鈭�}_{0}^{\sin 鈦� x} 鈥� \sqrt{t} d t) \\ = \frac{d}{d x} (x^{2} \sqrt{\sin 鈦� x} \cos 鈦� x) \\ = 2 x \sqrt{\sin 鈦� x} \cos 鈦� x + x^{2} (\frac{d}{d x} \sqrt{\sin 鈦� x} \cos 鈦� x) \\ = 2 x \sqrt{\sin 鈦� x} \cos 鈦� x + x^{2} (\frac{\cos 鈦� x}{2 \sqrt{\sin 鈦� x}} \cos 鈦� x + \sqrt{\sin 鈦� x} (鈭� \sin 鈦� x)) \\ = 2 x \sqrt{\sin 鈦� x} \cos 鈦� x + x^{2} \frac{\cos^{2} 鈦� x}{2 \sqrt{\sin 鈦� x}} 鈭� x^{2} \sin 鈦� x \sqrt{\sin 鈦� x} \end{aligned}$

Therefore, the answer is $2 x \sqrt{\sin 鈦� x} \cos 鈦� x + x^{2} \frac{\cos^{2} 鈦� x}{2 \sqrt{\sin 鈦� x}} 鈭� x^{2} \sin 鈦� x \sqrt{\sin 鈦� x}$

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Use the Second Fundamental Theorem of Calculus, if needed, to calculate each of the derivatives given below.
$\frac{d}{d x} (x^{2} {鈭�}_{0}^{\sin 鈦� x} 鈥� \sqrt{t} d t)$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The objective is to calculate the derivative.

Step 3. The derivate expression can be written as,

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91影视

Use the Second Fundamental Theorem of Calculus, if needed, to calculate each of the derivatives given below. ddxx2鈭�0sin鈦�x鈥�tdt

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The objective is to calculate the derivative.

Step 3. The derivate expression can be written as,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Logic and Functions

Theoretical and Mathematical Physics

Pure Maths

Discrete Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Use the Second Fundamental Theorem of Calculus, if needed, to calculate each of the derivatives given below.
$\frac{d}{d x} (x^{2} {鈭�}_{0}^{\sin 鈦� x} 鈥� \sqrt{t} d t)$