Q. 16 For each pair of functions ux聽a... [FREE SOLUTION]

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Chapter 5: Q. 16 (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) $鈭� u d v$ =
$u (x) = x$ , $v (x) = \cos 2 x$

Short Answer

Expert verified

The blank is

Part (a) $- 2 x \sin 2 x + \cos 2 x$

Part (b) $- 2 x \sin 2 x + \cos 2 x$

Part (c) $- 4 x \cos 2 x + 2 \sin 2 x$

Step by step solution

Part (a). Step 1. Given information

The given functions are $u (x) = x$ and $v (x) = \cos 2 x$ .

Part (a). Step 2. Evaluate ddxuxvx.

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

$\begin{array}{rcl} \frac{d}{d x} (u (x) v (x)) & = & \frac{d}{d x} (x \cos 2 x) \\ = & x \frac{d}{d x} \cos 2 x + \cos 2 x \frac{d}{d x} x \\ = & x (- 2 \sin 2 x) + \cos 2 x \\ = & - 2 x \sin 2 x + \cos 2 x \end{array}$

Part (b). Step 1. Integration

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

localid="1648820069229" $\begin{array}{rcl} 鈭� \frac{d}{d x} (u (x) v (x)) d x & = & 鈭� (- 2 x \sin 2 x + \cos 2 x) d x \\ u (x) v (x) + C & = & 鈭� (- 2 x \sin 2 x + \cos 2 x) d x \end{array}$

From the obtained equation, the integrand missing in part (b) is $- 2 x \sin 2 x + \cos 2 x$ .

Part (c). Step 1. Differentiation

Differentiate $v (x) = \cos 2 x$ with respect to x to obtain the value of dv.

$\begin{array}{rcl} \frac{d v}{d x} & = & \frac{d}{d x} \cos 2 x \\ = & - 2 \sin 2 x \\ d v & = & - 2 \sin 2 x d x \end{array}$

Part (c). Step 2. Integration

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

$\begin{array}{rcl} 鈭� u d v & = & 鈭� - 2 x \sin 2 x d x \\ = & - 2 鈭� x \sin 2 x d x \\ = & - 2 [x (2 \cos 2 x) - 鈭� (1) (2 \cos 2 x) d x] \\ = & - 2 [2 x \cos 2 x - 2 鈭� \cos 2 x d x] \\ = & - 2 [2 x \cos 2 x - 2 \frac{\sin 2 x}{2}] \\ = & - 4 x \cos 2 x + 2 \sin 2 x \end{array}$

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91影视

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) $鈭� u d v$ =
$u (x) = x$ , $v (x) = \cos 2 x$

Short Answer

Step by step solution

Part (a). Step 1. Given information

Part (a). Step 2. Evaluate ddxuxvx.

Part (b). Step 1. Integration

Part (c). Step 1. Differentiation

Part (c). Step 2. Integration

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91影视

For each pair of functions uxand vxin Exercises 16-18, fill in the blanks to complete each of the following:(a) ddxuxvx=______(b) 鈭�____dx=uxvx+C(c) 鈭�udv=______ux=x,vx=cos2x

Short Answer

Step by step solution

Part (a). Step 1. Given information

Part (a). Step 2. Evaluate ddxuxvx.

Part (b). Step 1. Integration

Part (c). Step 1. Differentiation

Part (c). Step 2. Integration

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

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Study anywhere. Anytime. Across all devices.

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) $鈭� u d v$ =
$u (x) = x$ , $v (x) = \cos 2 x$