Improper Integrals

An improper integral is one in which the integrand has one or more points of discontinuity, or at least one limit is infinite. In spite of this, the integration can often still be performed.
Example:  
\[\int^2_1 \frac{1}{\sqrt{x-1}} dx\]
.
The integrand is discontinuous at  
\[x=1\]
.
Substitute  
\[u=x-1\]
  then  
\[du=dx, \; x=1 \rightarrow u=0, \; x=2 \rightarrow u=1\]
  and the integral becomes  
\[\int^1_0 \frac{1}{u^{1/2}} du= \int^1_0 u^{-1/2} du=[\frac{u^{1/2}}{1/2}]^1_0= 2 \sqrt{1}-2(0)=2\]

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