I would appreciate advice on solving the following problem.

Evaluate the definite integral:

**integral (upper limit 2, lower limit 1) [ 2x (x^2 +1 )] dx**
Here is my own attempt.

Step 1: Solve for the generic (indefinite) integral via substitution

Let

**u = x^2 + 1**
So then

du / dx = 2x

2x dx = du

**dx = du / 2x**
Returning to the original function:

integral 2x (x^2 + 1) dx

= integral 2x (u) dx

= integral 2x (u) (du / 2x)

= integral u du

= integral x^2 + 1

=

**(x^3 / 3 ) + x + C**
Step 2: solve for the definite integral given the stated limits

At upper limit of 2:

((2)^3 / 3) + (2) +C

= (8/3) +2 + C

= (8/3) + (6/3) + C

**= 14/3 + C**
At lower limit of 1

((1)^3 / 3) + (1) +C

=(1/3) +1 + C

= (1/3) + (3/3) + C

**= 4/3 + C**
Subtracting the lower limit result from upper limit result:

(14 /3 + C) - (4/3 + C)

=14/3 + C - 4/3 - C

**= 10 / 3**
But my textbook and calculator both say the answer is

**(21/2)**
What am I doing wrong?