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For each integral in Exercises 5–8, write down three integrals that will have that form after a substitution of variables.

1udu

Short Answer

Expert verified

The three integrals 2x+1x2+xdx,sinxcosxdxand1xlnxdxwill have form 1uduafter a substitution of variables.

Step by step solution

01

Step 1. Given Information 

For given integral we have to write down three integrals that will have that form after a substitution of variables.
1udu

02

Step 2. The first integrals that will have that form after a substitution of variables.

Let

u=x2+xdudx=2x+1du=(2x+1)dx

This substitution changes the integral into

localid="1649155815093" 2x+1x2+xdx

03

Step 3. The first integrals that will have that form after a substitution of variables.

Let

u=cosxdudx=sinxdu=sinxdx

This substitution changes the integral into

role="math" localid="1648747414306" sinxcosxdx

04

Step 4. The first integrals that will have that form after a substitution of variables.

Let

u=lnxdudx=1xdu=1xdx

This substitution changes the integral into

1xlnxdx

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