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Provide a justification for each equality in the statement of the integration-by-parts formula for definite integrals from Theorem 5.10.

Short Answer

Expert verified

The definite integral of is a number and represents the area under the curve from a to b.

Step by step solution

01

Step 1. Given information

Theorem 5.10 is as follows:

Ifu=u(x),v=v(x)are differentiable functions ona,b, thenabudv=udvab=uvab-abvdu

02

Step 2. Explanation

According to theorem 5.10,

abudv=udvab=uvab-abvdu

Here, u(x),v(x)are differentiable functions on a,b

The definite integral of is a number and represents the area under the curve from a to b.

Take u,dvas first and second functions respectively.

So,abudv=udvab=uvab-abvdu

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Most popular questions from this chapter

Consider the integral sinxcosxdx.

(a) Solve this integral by using u-substitution with u=sinx and du=cosxdx.

(b) Solve the integral another way, using u-substitution with u=cosx and du=sinxdx.

(c) How must your two answers be related? Use algebra to prove this relationship.

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