Eat,Split,Substitute,Repeat

Calculus Level 4

∫ 0 n f ( x ) d x = 1729 \large\displaystyle\int _{ 0 }^{ n }{f( x )\, dx } =1729
Evaluate ∑ k = 1 n ∫ 0 1 f ( k − 1 + x ) d x \large\displaystyle\sum _{ k=1 }^{ n }{ \int _{ 0 }^{ 1 }{ f\left( k-1+x \right) dx } }


The answer is 1729.

Rohit Ner by Rohit Ner , 22, India

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1 solution

B B
Jan 21, 2016

Consider breaking up the domain of the function into the following intervals; [0,1], [1,2],...[n-1, n] Each part of the sum can be though of as translating the different sections of the domain (and the function) to [0,1] and then integrating. Hence, this gives the same as integrating the function between 0 and n.

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