Arithmetic slips evaluating a definite integral

Once the integration itself is done, attention drops, and that is exactly when the marks slip away. Evaluating a definite integral means computing $F(b)-F(a)$, the value of the antiderivative at the top limit minus its value at the bottom.

The most common slip is mishandling that subtraction: forgetting to substitute the lower limit at all, subtracting only part of $F(a)$, or, most often, dropping a sign when $F(a)$ is negative, so that a subtraction which should become an addition is written as an ordinary take-away. Fraction arithmetic adds another trap, since $F(b)$ and $F(a)$ are often fractions with different denominators that must be combined carefully.

None of it is hard mathematics; it is the quiet, low-attention stretch after the interesting part, which is precisely why it costs so many students their accuracy marks. The reassuring part is that the whole step is completely mechanical: once you settle a clear layout for the substitution, there is nothing left to judge, only to compute carefully.

Evaluate $\int_{-1}^{2}(2x+3)\,dx$. The antiderivative is $F(x)=x^{2}+3x$, so $F(2)=4+6=10$ and $F(-1)=1-3=-2$.

The tempting slip subtracts $2$ instead of $-2$:

Because $F(-1)=-2$, subtracting it flips to a plus. Write $F(a)$ in brackets so the double negative is visible:

1. Integrate first and write $F(x)$ clearly inside square brackets with the limits, and no $+c$, since it cancels in a definite integral.
2. Substitute the upper limit to get $F(b)$, writing the whole value inside its own brackets.
3. Substitute the lower limit to get $F(a)$, again inside brackets, keeping every sign.
4. Form the subtraction as $(F(b))-(F(a))$; if $F(a)$ is negative, the minus and the negative combine into a plus.
5. Simplify fractions carefully and, where sensible, glance at whether the size of the answer looks reasonable.

Our teachers treat the substitution step as a place to slow down, not speed up. In a one-to-one lesson we insist on brackets around each of $F(b)$ and $F(a)$, so a negative lower value cannot hide, and we drill the habit of reading a double negative aloud as a plus.

We also pair the work with the non-programmable scientific calculator you will use in the exam, checking each fraction as you go. Because the paper awards method marks, a clearly laid-out substitution keeps most of your credit even if one number slips.

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