Evaluate the following integrals as a limit of sums:


Formula used:



where,



Here, a = 0 and b = 2


Therefore,




Let,



Here, f(x) = x2 + 2x + 1 and a = 0




Now, by putting x = 0 in f(x) we get,


f(0) = 02 + 2(0) + 1 = 0 + 0 + 1 = 1


f(h)


= (h)2 + 2(h) + 1


= h2 + 2h + 1


Similarly, f(2h)


= (2h)2 + 2(2h) + 1




Since 1 is repeating n times in the series



Now take h2 and 2h common in remaining series





Put,



Since,
















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