Integration Limits Summation

Question

Show step by step how the integral, summation, and limit below equals 156.

 

Answer 1

Step 1:

The expression is .

Consider the left side part .

Step 1:

The expression is

Consider the left side part .

Since    

  

Step 2:

Consider the right hand part

Recall the algebraic formula 

Step 3:

Sum of natural numbers is .

Sum of squares of the natural numbers is .

Sum of cubes of natural numbers is .

 

Solution:

Hence the expression is verified.