Q. 39 Use the Second Fundamental Theor... [FREE SOLUTION]

Chapter 4: Q. 39 (page 399)

Use the Second Fundamental Theorem of Calculus, if needed, to calculate each of the derivatives given below.
$\frac{d}{d x} {鈭�}_{1}^{\sqrt{x}} 鈥� x \ln 鈦� t d t$

Short Answer

Expert verified

Ans: $\frac{d}{d x} {鈭�}_{1}^{\sqrt{x}} 鈥� x \ln 鈦� t d t = {鈭�}_{1}^{\sqrt{x}} 鈥� \ln 鈦� t + \frac{1}{2} \sqrt{x} \ln 鈦� \sqrt{x}$

Step by step solution

Step 1. Given information.

given,

$\frac{d}{d x} {鈭�}_{1}^{\sqrt{x}} 鈥� x \ln 鈦� t d t$

Step 2. The objective is to calculate the derivative.

Now, if $f$ is continuous on $[a, b]$ then for all $x 鈭� [a, b]$ ,

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

so,

$\begin{aligned} f (u (t)) = \ln 鈦� \sqrt{t} \\ f (u (x)) = \ln 鈦� \sqrt{x} \\ u^{鈥�} (x) = \frac{1}{2} \sqrt{x} \\ f (u (x)) u^{鈥�} (x) = \ln 鈦� \sqrt{x} \frac{1}{2} \sqrt{x} = \frac{1}{2} \sqrt{x} \ln 鈦� \sqrt{x} \end{aligned}$

Step 3. The derivative expression can be written as,

$\begin{aligned} \frac{d}{d x} {鈭�}_{1}^{\sqrt{x}} 鈥� x \ln 鈦� t d t \\ = {鈭�}_{1}^{\sqrt{x}} 鈥� \ln 鈦� t + \frac{1}{2} \sqrt{x} \ln 鈦� \sqrt{x} [f (u (x)) u^{鈥�} (x) = \frac{1}{2} \sqrt{x} \ln 鈦� \sqrt{x}] \end{aligned}$

Therefore, the answer is ${鈭�}_{1}^{\sqrt{x}} 鈥� \ln 鈦� t + \frac{1}{2} \sqrt{x} \ln 鈦� \sqrt{x} .$

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91影视

Use the Second Fundamental Theorem of Calculus, if needed, to calculate each of the derivatives given below.
$\frac{d}{d x} {鈭�}_{1}^{\sqrt{x}} 鈥� x \ln 鈦� t d t$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The objective is to calculate the derivative.

Step 3. The derivative 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. ddx鈭�1x鈥�xln鈦�tdt

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The objective is to calculate the derivative.

Step 3. The derivative expression can be written as,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Mechanics Maths

Discrete Mathematics

Calculus

Decision Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Use the Second Fundamental Theorem of Calculus, if needed, to calculate each of the derivatives given below.
$\frac{d}{d x} {鈭�}_{1}^{\sqrt{x}} 鈥� x \ln 鈦� t d t$