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Explain why 2xx2+1dxand 1xlnxdxare essentially the same integral after a change of variables.

Short Answer

Expert verified

Both integrals turn into 1uduafter a change of variables; u=x2+1in the first case, u=lnxin the second.

Step by step solution

01

Step 1. Given Information

Explain why 2xx2+1dxand role="math" localid="1648740843374" 1xlnxdxare essentially the same integral after a change of variables.

02

Step 2. Firstly changing the variable of ∫2xx2+1dx.

Let x2+1=u

u=x2+1dudx=2xdu=2xdx

This substitution changes the integral into

role="math" localid="1648740683170" 2xx2+1dx=1udu

03

Step 3. Now changing the integral ∫1xlnxdx.

Let lnx=u

u=lnxdudx=1xdu=1xdx

This substitution changes the integral into

localid="1648740902640" 1xlnxdx=1udu

Both integrals turn into 1uduafter a change of variables; u=x2+1 in the first case,u=lnx in the second.

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