Chapter 5: Q 89. (page 431)

Prove the integration formula.
$鈭� \ln x d x = x \ln x - x + c$
(a) by applying integration by parts to $鈭� \ln x d x$ .
(b) by differentiating $x \ln x - x$ .

Short Answer

Expert verified

(a) The solution is $鈭� \ln x d x = x \ln x - x + c$ .

(b) The solution is $\ln x$ .

Step by step solution

Part (a) Step 1. Given information.

It is given that $鈭� \ln x d x = x \ln x - x + c$ .

Part (a) Step 2. Prove the given formula by applying the integration by parts.

Take $u = \ln x$ and $v = 1$ .

$\begin{array}{rcl} 鈭� \ln x 路 1 d x & = & \ln x 鈭� d x - 鈭� [\frac{d}{d x} (\ln x) 鈭� d x] d x \\ = & x 路 \ln x - 鈭� [\frac{1}{x} 路 x] d x \\ = & x 路 \ln x - 鈭� d x \\ = & x 路 \ln x - x + C \end{array}$

Hence, the formula is proved.

Part (a) Step 1. Prove the formula by differentiating x lnx-x.

$\begin{array}{rcl} \frac{d}{d x} (x 路 \ln x - x) & = & \frac{d}{d x} (x 路 \ln x) - \frac{d}{d x} (x) \\ = & x \frac{d}{d x} (\ln x) + \ln x \frac{d}{d x} (x) - \frac{d}{d x} (x) \\ = & x 路 \frac{1}{x} + \ln x - 1 \\ = & 1 + \ln x - 1 \\ = & \ln x \end{array}$

Hence, the formula is proved.

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91影视

Prove the integration formula.
$鈭� \ln x d x = x \ln x - x + c$
(a) by applying integration by parts to $鈭� \ln x d x$ .
(b) by differentiating $x \ln x - x$ .

Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (a) Step 2. Prove the given formula by applying the integration by parts.

Part (a) Step 1. Prove the formula by differentiating x lnx-x.

One App. One Place for Learning.

Most popular questions from this chapter

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91影视

Prove the integration formula.鈭�lnxdx=xlnx-x+c(a) by applying integration by parts to 鈭�lnxdx.(b) by differentiatingxlnx-x.

Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (a) Step 2. Prove the given formula by applying the integration by parts.

Part (a) Step 1. Prove the formula by differentiating x lnx-x.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Decision Maths

Mechanics Maths

Probability and Statistics

Geometry

Pure Maths

Study anywhere. Anytime. Across all devices.

Prove the integration formula.
$鈭� \ln x d x = x \ln x - x + c$
(a) by applying integration by parts to $鈭� \ln x d x$ .
(b) by differentiating $x \ln x - x$ .