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Show that choosing a different anti-derivative v+Cof dvwill yield an equivalent formula for integration by parts,

as follows:

(a) Explain why what we need to show is that uv-vdu=uv+C-v+Cdu.

(b) Rewrite the equation from part (a), substitutingu=u(x), v=v(x), and du=u'(x)dx.

(c) Prove the equality you wrote down in part (b).

Short Answer

Expert verified

Part (a) We have added the constant term, which was eliminated during the derivative.

Part (b) uxvx-vxu'xdx=uxvx+C-vx+Cu'xdx

Part (c) Proved in the step.

Step by step solution

01

Part (a) Step 1. Explain the result.

Consider the formula of by part,

udv=uv-vdu

Substitute v=v+Cin the above integral.

uv-vdu=uv+C-v+Cdu

In the above by part method, by use v+C in place of v because we are doing integration and take the first function in terms of v. As the constant term C is added because the derivative of constant is zero.

02

Part 2. Step 1. Explanation.

Substitute the given values into the given equation.

uxvx-vxu'xdx=uxvx+C-vx+Cu'xdx

03

Part (c) Step 1. Explanation.

Prove the equality you wrote down in part (b).

let y be a function,

y=uv

Do the derivative of the functions.

dydx=duvdx=udvdx+vdudx

Rearrange the rule to solve it.

udvdx=duvdx-vdudx

Integrate both sides.

udvdxdx=duvdxdx-vdudxdxudv=uv-vdu

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Most popular questions from this chapter

Solve given definite integral.

1/41/21x21x2dx

Solvex21+x2dx the following two ways:

(a) with the substitution u=tan-1x;

(b) with the trigonometric substitution x = tan u.

Domains and ranges of inverse trigonometric functions: For each function that follows, (a) list the domain and range, (b) sketch a labeled graph, and (c) discuss the domains and ranges in the context of the unit circle.

f(x)=sec1x

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The substitution x = 2 sec u is a suitable choice for solving1x24dx.

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving1x24dx.

(c) True or False: The substitution x = 2 tan u is a suitable choice for solving1x2+4dx.

(d) True or False: The substitution x = 2 sin u is a suitable choice for solvingx2+45/2dx

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form x2a2.

(f) True or False: Trigonometric substitution doesn’t solve an integral; rather, it helps you rewrite integrals as ones that are easier to solve by other methods.

(g) True or False: When using trigonometric substitution with x=asinu, we must consider the cases x>a and x<-a separately.

(h) True or False: When using trigonometric substitution with x=asecu, we must consider the cases x>a and x<-a separately.

Solve each of the integrals in Exercises 39–74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.

9-x2xdx

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