How to solve the hardest integration questions in Cambridge A Level Math

Integration is one of the most heavily marked topics in Cambridge A Level Math, and in Paper 3 it is mostly a matter of choosing the right method quickly. Questions feel difficult mainly when one of the standard rules is missing from your toolkit. This guide follows the order of the P3 integration revision workshop on our channel: the Paper 1 foundation first, then direct integrals, fractions, products, substitution and powers of trigonometric functions, and finally how to choose between them.

Start from the Paper 1 foundation

Paper 3 builds on three Paper 1 ideas, and all three still appear.

• The power rule works for a linear expression only. For (mx + c)ⁿ, add one to the power, divide by the new power, and also divide by the derivative of the bracket, which is m. If the expression inside is not linear, the rule does not apply.
• A curve from its derivative. Integrate, add the constant, then use a point on the curve to find it.
• Area and volume formulas are not in the formula sheet. Learn the area and the volume of revolution about each axis. Volumes carry a pi outside, and for the y-axis versions the integrand is in terms of y and the limits are values of y.

Step 1: Know the direct integrals

Before any technique, check whether the integrand is already a standard result. The exponential, sine, cosine and squared secant integrals are in the formula sheet. Three more are not, but they are simply the differentiation results read backwards: secant times tangent, cosecant times cotangent, and squared cosecant. Look at the differentiation page of the formula sheet and reverse it. Whenever x is replaced by ax + b, divide by a.

Two habits catch marks here. A product can hide a direct integral: tan(2x) over cos(2x) is sec(2x) tan(2x), so no integration by parts is needed. And the rules need a linear expression: sin(x² + 2) cannot be integrated by dividing by a derivative.

Step 2: Fractions have a checklist

When the integrand is a fraction, work down this list in order.

• The ln rule. If the numerator is the derivative of the denominator, the integral is the natural log of the denominator. You may fix the numerator by multiplying and dividing by a constant, never by x. Tangent and cotangent are integrated this way: write them as sine over cosine and cosine over sine.
• The inverse tangent rule. A constant numerator over x² plus a constant. Make the coefficient of x² equal to one first, and write a non-square constant as a surd squared.
• Partial fractions. If the denominator factorises, split it. Sometimes you must factorise it yourself. A sum of two squares does not factorise. Two special difference-of-squares forms are given in the formula sheet, so check there before doing the full working.
• Split the numerator. A numerator such as a multiple of x plus a constant can be split into two fractions, one for the ln rule and one for the inverse tangent rule. You can split a numerator, never a denominator.
• Improper fractions. If the degree of the top is at least the degree of the bottom, divide first: quotient plus remainder over divisor.
• Move the denominator up. A power of a bracket in the denominator can become a negative power. A squared cosine in the denominator becomes a squared secant, which integrates directly.

Step 3: Products and integration by parts

Write the method as the integral of u times v′ equals uv minus the integral of u′ times v. The choice of which factor is u is the decision that matters. The order ILATE settles it: inverse trigonometric, logarithm, algebraic, trigonometric, exponential. Whichever comes first in that list is u. The reason is that v′ must be something you can integrate, and u should get simpler when differentiated. After each step, check that the new integral is simpler. A power of x that goes down is a good sign, and a power that goes up means the choice was wrong.

Three situations are worth knowing in advance:

• A power of x times an exponential may need parts twice.
• The integrals of ln x and inverse tangent x are found by parts, by writing the function as 1 times itself. Learn that they are done this way.
• With an exponential times a sine, or sine times cosine, the same integral may come back. Call it I, then solve the equation for I. This is uncommon, but it is worth recognising.

There is also a useful shortcut that is not named in the syllabus: if the derivative of the inner expression is multiplying outside, the integral of fⁿ times f′ is fⁿ⁺¹ over n + 1. It handles sin x cos x, and sin³x cos x, without a substitution.

Step 4: Substitution without losing marks

The substitution is always given, with one exception: the cubes of sine, cosine and tangent, covered in Step 5. The working has a fixed order:

1. Differentiate the substitution and make dx the subject.
2. Replace every part of the integral, including dx. If an x is left over, rearrange the substitution to write it in terms of u.
3. If the integral is definite, convert the limits at the same time. Write the lower limit and the upper limit separately, because the converted lower limit can turn out to be the larger number.
4. Integrate, then evaluate at the new limits.

If a “show that” question wants the larger limit on top, multiply the integrand by −1 and swap the limits. If a limit is infinity, integrate normally and then think about the graph: the inverse tangent of infinity is half of pi.

Step 5: Powers of sine, cosine and tangent

• Cubes. The three standard substitutions are u = cos x for sin³, u = sin x for cos³ and u = tan x for tan³. The question will not always give them. Alternatively, split off one factor and use the shortcut above.
• Even powers of sine and cosine. Use the double angle identity for cosine to remove the power. The angle on the right side is double the angle on the left, so sin²(2x) involves cos(4x).
• Tangent squared is secant squared minus one.
• Fourth powers need the identity twice. They usually come after an earlier part that proves an identity for you to use.

Step 6: Choosing the method

1. A single function, no product or fraction: a direct integral, possibly with a trigonometric identity.
2. A product: a given substitution if there is one. Otherwise integration by parts, or the shortcut if the derivative of the inner part is multiplying outside.
3. A fraction: work through the checklist in Step 2.
4. A power of a trigonometric function: odd powers by substitution, even powers by identity.

Use the earlier parts of the question

Many hard integrals are set up by an earlier part, even when the word “hence” is missing. An identity proved in part (a) is often what makes part (b) integrable. A product of sine and cosine with equal powers points to the double angle identity for sine. A derivative found in the first part tells you an integral in the second. If an integral looks impossible, reread the earlier parts.

Areas and volumes

Area questions in Paper 3 usually add a second layer of finding the limits yourself. Solve for the points of intersection, sketch the region, and decide which curve is on top. Volumes of revolution are in the syllabus but appear less often in Paper 3. Square the function before integrating, and remember that squaring a bracket gives three terms.

Exact answers

• Show the step where you substitute the limits. Method marks depend on it.
• Combine logarithms into a single term when the question asks for an exact answer in that form.
• Treat a letter such as a as a number. The answer may stay in terms of it, and the laws of logarithms, such as ln 3a = ln 3 + ln a, can remove it when the question says the final answer has no a.
• An answer that differs from the mark scheme may still be equivalent. Two correct integrals can differ by a constant, which is absorbed into the constant of integration.

Habits that protect the marks

• Differentiate your answer. On any indefinite integral you can check the result in under a minute.
• Do not forget the constant on indefinite integrals.
• Write brackets around negative values when substituting limits.
• Keep exact values until the final line, then round only if the question asks.
• Look at the marks. A short question for 2 marks has a one-step method. A long one for 7 or 8 marks is usually two techniques joined together.