Chapter 5: Q 89. (page 431)

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

Short Answer

Expert verified

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

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

Step by step solution

Part (a) Step 1. Given information.

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

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

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

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

Hence, the formula is proved.

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

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

Hence, the formula is proved.

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

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Prove the integration formula.∫lnxdx=xlnx-x+c(a) by applying integration by parts to ∫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

Discrete Mathematics

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Pure Maths

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Calculus

Study anywhere. Anytime. Across all devices.

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