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Chapter 5: Q. 17 (page 428)

For each pair of functions $u (x)$ and $v (x)$ in Exercises 16-18, fill in the blanks to complete each of the following:
(a) $\frac{d}{d x} (u (x) v (x))$ =_
(b) $鈭�d x = u (x) v (x) + C$
(c) $鈭� u d v =$
$u (x) = \ln x$ , $v (x) = x$

Short Answer

Expert verified

The blank is

Part (a) $1 + \ln x$

Part (b) $1 + \ln x$

Part (c) $x \ln x - x + C$

Step by step solution

01

Part (a). Step 1. Given information

The given functions are $u (x) = \ln x$ , $v (x) = x$ .

02

Part (a). Step 2. Evaluate ddxuxvx.

Differentiate the product of the given functions with respect to x by using integration by parts.

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

03

Part (b). Step 1. Integration

Integrate the obtained expression for $\frac{d}{d x} (u (x) v (x))$ with respect to x.

$\begin{array}{rcl} 鈭� \frac{d}{d x} (u (x) v (x)) d x & = & 鈭� (1 + \ln x) d x \\ u (x) v (x) + C & = & 鈭� (1 + \ln x) d x \end{array}$

04

Part (c). Step 1. Differentiation

Differentiate $v (x) = x$ with respect to x to obtain the value of dv.

$\begin{array}{rcl} \frac{d}{d x} v (x) & = & \frac{d}{d x} x \\ \frac{d v}{d x} & = & 1 \\ d v & = & d x \end{array}$

05

Part (c). Step 2. Integration

Substitute the given and obtained values to evaluate $鈭� u d v$ .

$\begin{array}{rcl} 鈭� u d v & = & 鈭� \ln x d x \\ = & 鈭� \ln x 路 1 d x \\ = & \ln x 路 x - 鈭� \frac{1}{x} x d x \\ = & x \ln x - x + C \end{array}$

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Most popular questions from this chapter

Solve the integral: $鈭� x \ln \sqrt{x} d x$

Give an example of an integral for which trigonometric substitution is possible but an easier method is available. Then give an example of an integral that we still don鈥檛 know how to solve given the techniques we know at this point.

Solve given definite integral.

${鈭�}_{1}^{2} 鈥� \frac{1}{x \sqrt{9 鈭� x^{2}}} d x$

Solve the integral: $鈭� \ln \sqrt{x} d x$

Solve the integral: $鈭� x \sin {(x)}^{2} d x$ .

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