Integration: choosing the method before you start

Look at it in this order

Before you write anything, run down this list. It takes ten seconds and it will save you from three-quarters of the dead ends.

Most integrals you meet are the first or second item. Recognising that quickly is worth more than being fast at the algebra afterwards.

• Is it a standard form you already know? Powers, exponentials, the basic trigonometric ones, and one over x.
• Is one part of it the derivative of another part? That is the reverse chain rule, and it is the single most common case.
• Can algebra make it standard? Expanding a bracket, splitting a fraction term by term, or simplifying a quotient before integrating - never after.
• Is there a trigonometric identity that turns it into something standard? Squared sines and cosines almost always want the double-angle identity first.
• Is a substitution suggested or sensible? If the question offers one, use it; if it does not, look for the inner function whose derivative is already sitting there.

Spotting the reverse chain rule

Differentiating a composite function multiplies by the derivative of the inside. So integrating something that already has that derivative sitting beside it undoes the whole operation at once.

The tell is a bracket or a function raised to a power, or inside an exponential or a logarithm, with something that looks like its derivative multiplying it - a linear term next to a quadratic inside a bracket, for example. When you see that pattern, name the inside function, differentiate it in the margin, and check whether what remains is a constant multiple of it. If it is, you are done in one line; the constant is just carried along.

If what remains is not a constant multiple of the derivative, stop. It is not this method, and forcing it is how people end up with an answer that differentiates back into something else entirely.

The check that takes ten seconds

Differentiate your answer. This is not optional advice for careful students; it is the only way to be certain, and it is quicker than re-reading your own working.

It catches the errors that are otherwise invisible: a missing constant factor, a sign dropped somewhere in the middle, a chain rule that was not fully undone. If differentiating your answer does not give you back exactly what you were asked to integrate, you know immediately, and you know before the marker does.

Do it on indefinite integrals every time. On definite integrals, differentiate the antiderivative before you substitute the limits - it is the same check and it costs nothing.

Areas, and the two sign traps

A definite integral is a signed area. Area below the horizontal axis contributes negatively, so an integral across a curve that crosses the axis gives you the difference between the regions, not the total area between the curve and the axis.

That is the first trap, and the fix is always the same: find where the curve crosses the axis, split the integral at those points, and take the absolute value of each piece before adding. A question that asks for the area of the region bounded by a curve and the axis wants the total; a question that asks for the value of the integral wants the signed answer. Read which one you were asked.

The second trap is the area between two curves. Subtract the lower function from the upper one, and be sure about which is which over the interval you are integrating - if they cross, the answer changes at the crossing point and you have to split there too. Sketching the two curves, however roughly, takes less time than recovering from getting this backwards.

When you are asked to approximate

Approximation questions are not asking you to integrate at all, and every year some students integrate anyway. The trapezoidal rule wants function values at evenly spaced points, and the marks are for setting it out correctly and using the right number of intervals - not for the arithmetic.

Two things to say when asked to comment on the result. Whether the estimate is above or below the true value depends on the concavity of the curve over that interval: a curve bending upwards is overestimated by straight chords, and one bending downwards is underestimated. And more intervals gives a better estimate, because the chords hug the curve more closely. Either observation is usually worth a mark and neither requires any calculation.

Where it usually goes wrong

These five account for most of the marks lost in integration questions, and none of them is about being bad at integrating.

• Starting to integrate before deciding what kind of integral it is.
• Omitting the constant of integration on an indefinite integral, which is a mark every time.
• Treating a signed integral as an area when the curve crosses the axis.
• Simplifying after integrating rather than before. Algebra first, always.
• Not differentiating the answer to check, on a question where the check would have taken ten seconds.

What to practise next

Take twenty integrals from anywhere and do not integrate any of them. Just write, next to each, which of the five categories above it belongs to. Sorting is the skill that is actually being tested in a mixed exercise, and it is the one nobody practises deliberately.

Then do ten area questions and sketch every one before integrating. You will find that the sketch decides the method, the limits and the sign, which means the integration was never the hard part.

Check yourself

Three questions on what is above. Have a go before you open them - reading an answer you have not tried to give is the version of this that does nothing.