   Chapter 13.6, Problem 1CP ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# True or false: In evaluating ∫ u   d v by parts,(a) the parts u and dv are selected and the parts du and v are calculated.(b) the differential (often dx) is always chosen as part of dv.(c) the parts du and v are found from u and dv as follows: d u = u '   d x  and  v   =   ∫ d v (d) For ∫ 3 x e 2 x we could choose u = 3 x and d v = e 2 x d x .

(a)

To determine

Whether the provided statement “In evaluating udv parts, the parts u and dv are selected and the parts du and v are calculated”.

Explanation

Integration by parts is an integration technique that uses a formula that follows from the product rule of derivatives,

ddx(uv)=udvdx+vdudx

On integrating this formula by rearranging the differentials, the formula for integration by parts is obtained as,

udv=uvvdu

Clearly, in the formula, the terms v and du are present, so yes, the parts u and dv are selected and the parts du and v are calculated.

For example, consider the integral,

1elnxdx

Now, split the integrand into two parts setting one part equal to u and another part equal to dv.

So, the values will be:

u=lnx

And,

dv=dx

Then,

du=1xdx

And,

v=

(b)

To determine

Whether the provided statement “In evaluating udv by parts, the differential (often dx) is always chosen as a part of dv”.

(c)

To determine

Whether the provided statement “In evaluating udv by parts, the parts du and v are found from u and dv as du=udx and v=dv.”

(d)

To determine

Whether the provided statement “For 3xe2xdx, the parts chosen can be u=3x and dv=e2xdx.”

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