Question

Use the form of the definition of the integral given in Theorem 4 to evaluate the integral.

Answer 1

Step 1:

The integral expression is .

Using theorem :

If is integrable on , then .

Where .

The integral expression is .

Here  and .

Substitute in .

.

Step 2:

                       ( Since )

              

Sum of first natural numbers is .

Sum of the squares of first natural numbers is .

Simplify the expression.

              

.

Solution :

.