设f(x)={1+x^2 (x=0) ,求∫(1,3) f(x-2)dx

题目

设f(x)={1+x^2 (x=0) ,求∫(1,3) f(x-2)dx

答案

取t=x-2,
∫(1,3) f(x-2)dx=∫(-1,1) f(t)dt
=∫(-1,0) f(t)dt+∫(0,1) f(t)dt
=∫(-1,0)[1+t^2]dt+∫(0,1)[e^t]dt
=e+1/3