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) $\intd x = u (x) v (x) + C$
(c) $\int u d v$ =
$u (x) = x$ , $v (x) = \cos 2 x$

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

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 $\int u d v$ .

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

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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) $\intd x = u (x) v (x) + C$
(c) $\int u d v$ =
$u (x) = x$ , $v (x) = \cos 2 x$

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Applied Mathematics

Decision Maths

Geometry

Discrete Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

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

Logic and Functions

Applied Mathematics

Decision Maths

Geometry

Discrete Mathematics

Probability and Statistics

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) $\intd x = u (x) v (x) + C$
(c) $\int u d v$ =
$u (x) = x$ , $v (x) = \cos 2 x$