Find INT{2,1}{x^4 + 3x^2 + 2}

This is a typical AS-level Maths question.

1. We see it’s a definite integral so start by drawing some big square brackets with the “limits” (2 and 1).

2. We go through the expression inside the integral (the “integrand”) term by term and integrate each one. To integrate, we raise the power by one and divide by the new power. E.g. “x to the 4” would become “ one fifth x to the five” [would use whiteboard here]

3. Remember, this is a definite integral so we don’t need to add ‘C’ at the end!

4. We now substitute x = 2 into our integrated expression (just go through and replace x’s with 2’s - don’t change anything else yet). Then we substitute x = 1 into the expression and subtract it from what we just worked out. [again, on whiteboard]

5. To make life easier, we keep the fractions as fractions and whole numbers as whole numbers until the end. We expand out the brackets carefully and, finally, combine everything into one number to get our final answer.

6. [As an extra] what we just worked out is the area underneath the “integrand” curve between x = 1 and x = 2.