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Use the chain rule and the Fundamental Theorem of Calculus to prove the integration-by-substitution formula for definite integrals:

โˆซabf'(u(x))u'(x)dx=f(u(b))โˆ’f(u(a))

Short Answer

Expert verified

After using the chain rule and the Fundamental Theorem of Calculus by the integration-by-substitution formula for definite integrals we proved thatโˆซabf'(u(x))u'(x)dx=f(u(b))โˆ’f(u(a))

Step by step solution

01

Step 1. Given Information 

Use the chain rule and the Fundamental Theorem of Calculus to prove the integration-by-substitution formula for definite integrals:

โˆซabf'(u(x))u'(x)dx=f(u(b))โˆ’f(u(a))

02

Step 2. To solve taking the left hand side integral. 

y=โˆซabf'(u(x))u'(x)dx

Let

t=u(x)dtdx=u'(x)dt=u'(x)dx

03

Step 3. Now the integral after substitution.

โˆซabf'(u(x))u'(x)dx=โˆซabf'(t)dtโˆซabf'(u(x))u'(x)dx=f(t)abโˆซabf'(u(x))u'(x)dx=f(u(x))abโˆซabf'(u(x))u'(x)dx=f(u(b))-f(u(a))

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