Thursday, August 15, 2019

calculus - Simpler way to compute a definite integral without resorting to partial fractions?


I found the method of partial fractions very laborious to solve this definite integral : 03x1+x2dx


Is there a simpler way to do this ?


Answer



Perhaps this is simpler.



Make the substitution x2/3=t. Giving us


2x1/33x2/3dx=dt, i.e x1/3dx=32tdt


This gives us that the integral is


I=320t1+t3 dt


Now make the substitution t=1z to get


I=32011+t3 dt


Add them up, cancel the 1+t, write the denominator (t2t+1) as (t+a)2+b2 and get the answer.


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