121616949-math.182

# 121616949-math.182 - x 2 x sin x 2 cos x C Such repeated...

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168 Chapter 8 Techniques of Integration At first this looks useless—we’re right back to R sec 3 x dx . But looking more closely: Z sec 3 x dx = sec x tan x - Z sec 3 x dx + Z sec x dx Z sec 3 x dx + Z sec 3 x dx = sec x tan x + Z sec x dx 2 Z sec 3 x dx = sec x tan x + Z sec x dx Z sec 3 x dx = sec x tan x 2 + 1 2 Z sec x dx = sec x tan x 2 + ln | sec x + tan x | 2 + C. EXAMPLE 8.14 Evaluate Z x 2 sin x dx . Let u = x 2 , dv = sin x dx ; then du = 2 x dx and v = - cos x . Now Z x 2 sin x dx = - x 2 cos x + Z 2 x cos x dx . This is better than the original integral, but we need to do integration by parts again. Let u = 2 x , dv = cos x dx ; then du = 2 and v = sin x , and Z x 2 sin x dx = - x 2 cos x + Z 2 x cos x dx = - x 2 cos x + 2 x sin x - Z 2 sin x dx = - x
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Unformatted text preview: x + 2 x sin x + 2 cos x + C. Such repeated use of integration by parts is fairly common, but it can be a bit tedious to accomplish, and it is easy to make errors, especially sign errors involving the subtraction in the formula. There is a nice tabular method to accomplish the calculation that minimizes the chance for error and speeds up the whole process. We illustrate with the previous example. Here is the table:...
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