H: Integration — AQA A-Level Mathematics
Test yourself on H: Integration with AQA A-Level practice questions.
7 days Premium · Then free forever · No card, no charge
H: Integration explained
The Fundamental Theorem of Calculus (FTC) links differentiation and integration.
Read the full explanation
It has two parts: (1) if F(x) = ∫_a^x f(t) dt, then F'(x) = f(x); (2) ∫_a^b f(x) dx = F(b) - F(a), where F is any antiderivative of f. This means differentiation and integration are inverse processes. To use it, find an antiderivative of the integrand, then evaluate it at the upper and lower limits and subtract. For example, ∫_0^2 x² dx = [x³/3]_0^2 = 8/3 - 0 = 8/3. The theorem also allows you to differentiate an integral with a variable limit. In exams, you may be asked to evaluate a definite integral, find a derivative of an accumulation function, or prove a result using the FTC. Understanding the theorem is essential for solving problems involving areas under curves and accumulated change.
Integrate xⁿ (excluding n = −1), and related sums, differences and constant multiples. Integrate e^(kx), 1/x, sin kx, cos kx and related sums, differences and constant multiples.
This section builds the core toolkit for indefinite integration. For powers, you reverse differentiation: raise the index by one and divide by the new index, so ∫xⁿ dx = xⁿ⁺¹/(n+1) + c, valid for any constant n except n = −1. The exception is handled by ∫(1/x) dx = ln|x| + c. You also integrate exponential and trigonometric functions: ∫e^(kx) dx = (1/k)e^(kx) + c, ∫sin kx dx = −(1/k)cos kx + c, and ∫cos kx dx = (1/k)sin kx + c. These rules extend to sums, differences and constant multiples by integrating term by term and taking constants outside the integral. Always include the constant of integration for indefinite integrals. For example, ∫(3x² + 2e^(2x) − 5/x) dx = x³ + e^(2x) − 5 ln|x| + c.
Evaluate definite integrals; use a definite integral to find the area under a curve and the area between two curves.
A definite integral has limits and gives a numerical value. To evaluate ∫ from a to b of f(x) dx, find an antiderivative F(x), then compute F(b) − F(a). The constant of integration cancels in the subtraction, so it is not needed for a definite integral. Geometrically, if f(x) ≥ 0 on [a, b], the integral equals the area under y = f(x) between x = a and x = b. If the curve dips below the x-axis, the integral gives a signed area; for total area, split the interval at roots and take absolute values. For the area between y = f(x) and y = g(x) from x = a to x = b, integrate upper minus lower: ∫(f(x) − g(x)) dx. Find intersection points to determine limits if not given. Example: area between y = x and y = x² from x = 0 to 1 is ∫(x − x²) dx = [x²/2 − x³/3] from 0 to 1 = 1/2 − 1/3 = 1/6.
Understand and use integration as the limit of a sum.
Definite integration is defined as the limit of a sum of areas of rectangles. For a function f(x) on [a, b], divide the interval into n strips of width Δx = (b−a)/n. The sum of areas of rectangles, Σ f(x_i) Δx, approximates the area under the curve. As n → ∞ and Δx → 0, the sum tends to the definite integral ∫_a^b f(x) dx. This limit exists for continuous functions (and piecewise continuous). The notation ∫ comes from an elongated S for sum. For example, to approximate ∫_0^1 x^2 dx, use right endpoints: Σ_{i=1}^n (i/n)^2 (1/n) = (1/n^3) Σ i^2 = (1/n^3) n(n+1)(2n+1)/6 → 1/3 as n → ∞. This shows the integral equals 1/3. Understanding this limit underpins the Fundamental Theorem of Calculus and allows numerical methods like the trapezium rule to be seen as finite approximations.
Carry out simple cases of integration by substitution and integration by parts; understand these methods as the inverse processes of the chain and product rules respectively. (Integration by substitution includes finding a suitable substitution and is limited to cases where one substitution will lead to a function which can be integrated; integration by parts includes more than one application of the method but excludes reduction formulae).
Integration by substitution reverses the chain rule. For ∫ f(g(x))g'(x) dx, let u = g(x), then du = g'(x) dx, giving ∫ f(u) du. For definite integrals, change limits. Example: ∫ 2x cos(x²) dx, let u = x², du = 2x dx, giving ∫ cos u du = sin u + C = sin(x²) + C. Integration by parts reverses the product rule: ∫ u dv/dx dx = uv − ∫ v du/dx dx. Choose u and dv wisely (e.g., u = x, dv = e^x dx). Example: ∫ x e^x dx = x e^x − ∫ e^x dx = x e^x − e^x + C. Integration by parts may need more than one application, e.g. ∫ x² e^x dx requires two applications; reduction formulae are excluded. Some integrals reappear after applying by parts and are solved algebraically.
Integrate using partial fractions that are linear in the denominator.
This topic teaches you to integrate rational expressions by first splitting them into partial fractions with linear denominators. For example, to integrate (2x+3)/((x+1)(x-2)), you write it as A/(x+1) + B/(x-2), solve for A and B, then integrate each term as A ln|x+1| + B ln|x-2| + C. The method requires the degree of the numerator to be less than the denominator; if not, perform polynomial division first. You must also handle repeated linear factors, such as (x+1)^2, by including terms A/(x+1) + B/(x+1)^2. The skill is assessed in problems that require algebraic manipulation, correct integration of 1/(ax+b), and appropriate use of modulus signs and the constant of integration.
Evaluate the analytical solution of simple first order differential equations with separable variables, including finding particular solutions (Separation of variables may require factorisation involving a common factor).
This topic covers solving first-order differential equations where the variables can be separated. You rearrange the equation so that all terms involving y are on one side and all terms involving x on the other, then integrate both sides. For example, dy/dx = xy can be written as (1/y) dy = x dx, leading to ln|y| = x^2/2 + C. If an initial condition is given, substitute to find the particular solution. Sometimes factorisation is needed first, such as dy/dx = x(y+1) becoming (1/(y+1)) dy = x dx. The solution may be left in implicit form or rearranged to make y the subject. Assessment focuses on correct separation, integration, and use of initial conditions.
Interpret the solution of a differential equation in the context of solving a problem, including identifying limitations of the solution; includes links to kinematics.
When you solve a differential equation, the general solution contains an arbitrary constant; applying an initial or boundary condition fixes it. Interpreting the solution means explaining what the mathematical expression represents in the original problem, including units and behaviour over time. For example, in kinematics, solving dv/dt = −kv with v(0) = u gives v = ue^(−kt), so speed decays exponentially. You must also identify limitations: the model may predict unrealistic values, such as negative speed or unbounded growth, or may only be valid for a restricted time interval. Checking the domain and physical constraints is part of the interpretation.
Your focus
- State the Fundamental Theorem of Calculus and explain its two parts.
- Evaluate definite integrals using antiderivatives and limits.
- Differentiate integrals with variable limits, including cases requiring the chain rule.
Show all 25 objectives
- Apply the power rule to integrate xⁿ for n ≠ −1, including fractional and negative indices.
- Integrate e^(kx), 1/x, sin kx and cos kx correctly, including the necessary reciprocal factors and signs.
- Integrate sums, differences and constant multiples of these functions term by term, and include the constant of integration.
- Evaluate definite integrals using the fundamental theorem of calculus, applying limits correctly.
- Calculate the area under a curve using a definite integral, including handling regions below the x-axis.
- Determine the area between two curves by integrating the difference between the upper and lower functions over the correct interval.
- Explain how a definite integral is defined as the limit of a sum of areas of rectangles.
- Calculate the limit of a sum for a simple polynomial function over a given interval.
- Interpret the limit of a sum as the signed area under a curve and relate it to the definite integral notation.
- Perform integration by substitution for simple cases, including finding a suitable substitution and changing limits for definite integrals.
- Apply integration by parts, including repeated application, to integrate products of functions.
- Explain how integration by substitution and integration by parts are the inverse processes of the chain rule and product rule respectively.
- Decompose a rational function with linear denominators into partial fractions.
- Integrate expressions of the form 1/(ax+b) correctly.
- Apply the method to definite and indefinite integrals, including those requiring polynomial division.
- Separate variables in a first-order differential equation.
- Integrate both sides to find the general solution.
- Use an initial condition to determine a particular solution.
- Factorise expressions to enable separation of variables.
- Solve a first-order differential equation and apply an initial condition to find a particular solution.
- Explain the meaning of the solution in the context of the original problem, including units and behaviour.
- Evaluate the limitations of a differential equation model in a real-world context, such as kinematics.
H: Integration exam tips
Marking Points
- State the theorem correctly: differentiation and integration are inverse operations.
- Use the theorem to evaluate definite integrals by finding an antiderivative and applying limits.
- Differentiate an integral with a variable upper limit correctly.
- Apply the theorem to problems involving area under a curve or accumulated change.
- Recognise when the theorem is applicable and when adjustments (e.g., chain rule) are needed.
- Correctly applies the power rule ∫xⁿ dx = xⁿ⁺¹/(n+1) + c for n ≠ −1, including fractional and negative indices.
- Recognises that ∫(1/x) dx = ln|x| + c and does not incorrectly apply the power rule to n = −1.
- Integrates exponential functions using ∫e^(kx) dx = (1/k)e^(kx) + c, including the reciprocal factor 1/k.
- Integrates trigonometric functions with correct signs and reciprocal factors: ∫sin kx dx = −(1/k)cos kx + c and ∫cos kx dx = (1/k)sin kx + c.
- Handles sums, differences and constant multiples by integrating term by term and factoring out constants.
- Includes the constant of integration, c, for indefinite integrals.
- Evaluates a definite integral correctly by finding an antiderivative and substituting the upper and lower limits, then subtracting: F(b) − F(a).
- Uses square brackets to show the antiderivative evaluated at the limits, e.g. [F(x)] from a to b, before substituting.
- Interprets a definite integral as the area under a curve when the function is non-negative on the interval.
- Handles areas below the x-axis by splitting the interval at roots and taking absolute values to find total area.
- Finds the area between two curves by integrating (upper curve − lower curve) over the appropriate interval.
- Determines intersection points of two curves to find the limits of integration when they are not given.
- Defines the definite integral as the limit of a sum of the form lim_{n→∞} Σ_{i=1}^n f(x_i) Δx, where Δx = (b−a)/n and x_i are sample points in each subinterval.
- Explains that the limit is taken as the number of subintervals increases without bound and the width of each subinterval tends to zero.
- Applies the definition to a simple function, such as f(x) = x or f(x) = x^2, by setting up the sum and evaluating the limit.
- Interprets the limit as the signed area under the curve y = f(x) between x = a and x = b, noting that areas below the x-axis contribute negatively.
- Connects the limit definition to the notation ∫_a^b f(x) dx and recognises that the integral is independent of the choice of sample points for continuous functions.
- Selects an appropriate substitution u = g(x) such that the integral transforms into a standard form, and correctly computes du = g'(x) dx.
- Changes the limits of integration when evaluating a definite integral by substitution, or substitutes back to the original variable for indefinite integrals.
- Applies integration by parts using the formula ∫ u dv = uv − ∫ v du, choosing u and dv to simplify the resulting integral.
- Performs integration by parts more than once when necessary, for example for integrals involving x² e^x or x² sin x, until the integral is manageable.
- Recognises that integration by substitution is the inverse of the chain rule and integration by parts is the inverse of the product rule, and explains this relationship.
- Decompose the rational function into partial fractions with linear denominators, including cases with repeated factors.
- Integrate each partial fraction term correctly, using the standard result ∫ 1/(ax+b) dx = (1/a) ln|ax+b| + C.
- Include the constant of integration and use modulus signs inside logarithms where appropriate.
- If the numerator degree is not less than the denominator, perform polynomial division before decomposing.
- Combine logarithmic terms correctly, applying logarithm laws where simplification is required.
- Separate the variables correctly, ensuring all y terms are on one side and all x terms on the other.
- Integrate both sides with respect to their respective variables, including the constant of integration.
- Apply the initial condition to determine the value of the constant and obtain the particular solution.
- Rearrange the solution to the required form, such as y = f(x), if specified.
- Factorise expressions where necessary to enable separation, e.g., common factors.
- Correctly solves the differential equation to obtain a general solution, showing separation of variables as appropriate.
- Applies the given initial or boundary condition to determine the constant of integration and state the particular solution.
- Translates the mathematical solution back into the problem context, including correct units and an explanation of each variable.
- Identifies at least one limitation of the model, such as a prediction of negative speed, infinite displacement, or validity only for a certain range of time.
- Uses the solution to answer a specific question, for example finding the time when speed reaches a given value or the limiting speed.
- Comments on the reasonableness of the solution in the physical situation, linking the mathematical behaviour to the assumptions of the model.
Examiner Tips
- 💡Write the antiderivative in square brackets with limits to show your method clearly.
- 💡Check your answer by differentiating the antiderivative to see if you get the original integrand.
- 💡For variable limits, identify the inner function and apply the chain rule correctly.
- 💡Write the constant of integration, c, clearly in every indefinite integral answer; it is often worth a mark.
- 💡For terms like 1/x, rewrite as x⁻¹ only if you remember the exception; otherwise go straight to ln|x|.
- 💡Check your answer by differentiating it back to the original integrand; this catches sign and factor errors quickly.
- 💡When integrating sums, split the integral into separate terms and deal with each one carefully, especially signs.
- 💡Write the antiderivative in square brackets with the limits clearly shown before substituting; this helps you avoid sign errors.
- 💡If a curve crosses the x-axis in the interval, split the integral at the root(s) and evaluate each part separately, taking absolute values for total area.
- 💡For area between curves, sketch the graphs to identify which is upper and which is lower, and to see intersection points.
- 💡Check whether the question asks for 'area' or 'value of the integral'; if it asks for area, ensure your answer is positive.
- 💡When asked to 'use integration as the limit of a sum', set up the sum clearly, stating Δx and the sample points, then evaluate the limit using standard summation formulae.
- 💡Show all steps in evaluating the limit, including algebraic manipulation of sums such as Σ i and Σ i^2, and state the limiting value explicitly.
- 💡If a question asks for an interpretation, relate the limit to the area under the curve and mention that the integral gives the exact area as the number of strips tends to infinity.
- 💡When using substitution, clearly state u and du, and show the transformed integral before integrating.
- 💡For integration by parts, write down u, dv, du and v explicitly, then apply the formula step by step.
- 💡If a question asks for 'more than one application', be prepared to apply integration by parts repeatedly, simplifying each time, and watch for the integral reappearing (then solve algebraically).
- 💡Always check whether the rational function is proper; if not, perform polynomial division first.
- 💡When solving for constants, substitute convenient values of x that make denominators zero to simplify equations.
- 💡After integrating, simplify logarithmic expressions using log laws if the question asks for a single logarithm.
- 💡Remember to include the constant of integration in indefinite integrals; it is often required for full marks.
- 💡Always separate variables before integrating; do not integrate terms that are multiplied together.
- 💡When finding a particular solution, substitute the initial condition as soon as possible to find C.
- 💡If the question asks for y in terms of x, rearrange carefully, especially when logarithms or exponentials are involved.
- 💡Check your solution by differentiating to see if it satisfies the original differential equation.
- 💡Read the problem carefully to identify the dependent and independent variables, and note any given initial or boundary conditions before starting to solve.
- 💡Show all steps of the solution, including the separation of variables, and clearly state the general solution before applying conditions.
- 💡When interpreting, write a sentence explaining what the solution means in context, and explicitly mention any limitations such as the model breaking down for large t.
- 💡Check whether the question asks for a specific value or a general interpretation; answer precisely and include units where appropriate.
Common Mistakes
- Forgetting to subtract the value at the lower limit: always compute F(b) - F(a), not just F(b).
- Incorrect antiderivative: ensure you differentiate back to check. For example, the antiderivative of x² is x³/3, not x³/2.
- Misapplying the theorem when the upper limit is a function of x: then you must use the chain rule, e.g., d/dx ∫_a^{g(x)} f(t) dt = f(g(x))g'(x).
- Forgetting to divide by the new index when integrating powers, e.g. writing ∫x³ dx = x⁴ + c instead of x⁴/4 + c. Correction: always divide by (n+1).
- Applying the power rule to 1/x, giving x⁰/0, which is undefined. Correction: use ln|x| for the integral of 1/x.
- Omitting the constant of integration, c, in indefinite integrals. Correction: always add c unless evaluating a definite integral.
- Incorrect sign for ∫sin kx dx, writing +(1/k)cos kx instead of −(1/k)cos kx. Correction: remember the derivative of cos is −sin, so the integral of sin is −cos.
- Forgetting the reciprocal factor 1/k when integrating e^(kx), sin kx or cos kx. Correction: always divide by the coefficient of x.
- Subtracting the limits in the wrong order, e.g. F(a) − F(b) instead of F(b) − F(a). Correction: always do upper limit minus lower limit.
- Thinking that omitting the constant of integration in a definite integral is an error. Correction: the constant cancels in F(b) − F(a), so it is not needed for a definite integral; it is only required for an indefinite integral.
- Assuming the definite integral always gives the geometric area when the curve crosses the x-axis. Correction: split the interval at roots and take absolute values for total area.
- Integrating (lower curve − upper curve) when finding area between curves, giving a negative result. Correction: always subtract the lower curve from the upper curve.
- Not finding intersection points when limits are not given, leading to incorrect limits. Correction: set the curves equal and solve for x to find limits.
- Error: Treating Δx as a fixed small number rather than a quantity that tends to zero. Correction: Emphasise that Δx = (b−a)/n and as n → ∞, Δx → 0.
- Error: Using only left or right endpoints without considering the limit. Correction: Show that for continuous functions, any choice of sample points gives the same limit, and practise with different sample points.
- Error: Forgetting that the sum approximates signed area, so regions below the axis give negative contributions. Correction: Illustrate with a function that crosses the x-axis and compute the sum to see negative terms.
- Error: Assuming the limit always exists for any function. Correction: Note that continuity (or piecewise continuity) is sufficient for the limit to exist; discuss a counterexample like the Dirichlet function if appropriate.
- Forgetting to change the limits when using substitution in a definite integral. Correction: either convert the limits to u-values or substitute back before evaluating.
- Choosing u and dv poorly in integration by parts, leading to a more complicated integral. Correction: use the guideline of choosing u as a function that simplifies when differentiated (e.g. ln x, x, x²) and dv as something easily integrated (e.g. e^x, sin x).
- Omitting the constant of integration for indefinite integrals. Correction: always add + C for indefinite integrals.
- Attempting to use substitution when the derivative of the inner function is not present (up to a constant factor). Correction: check that the integrand contains g'(x) or a multiple of it; otherwise, the substitution may not work.
- Forgetting to divide by the coefficient of x when integrating 1/(ax+b). Correction: always write (1/a) ln|ax+b|.
- Omitting the modulus signs in logarithmic terms. Correction: use ln|ax+b| because the argument of a logarithm must be positive.
- Attempting partial fractions when the numerator degree is greater than or equal to the denominator. Correction: first divide the polynomials to obtain a proper fraction.
- Incorrectly setting up partial fractions for repeated linear factors, e.g., using only A/(x+1) for (x+1)^2. Correction: include both A/(x+1) and B/(x+1)^2.
- Forgetting to include the constant of integration after integrating. Correction: always add C immediately after integration.
- Incorrectly separating variables, e.g., not dividing both sides by the function of y. Correction: ensure the equation is in the form f(y) dy = g(x) dx.
- Misapplying the initial condition, such as substituting into the wrong side. Correction: substitute the given x and y values into the integrated equation to solve for C.
- Not factorising before separating, leading to an inseparable equation. Correction: look for common factors and factorise first.
- Forgetting to include the constant of integration and therefore losing the ability to apply initial conditions; always write + c after integrating.
- Applying the initial condition before finding the general solution, which leads to an incorrect constant; first integrate, then substitute.
- Ignoring the physical context and giving a purely algebraic answer without units or interpretation; always relate variables to the problem and state units.
- Assuming the model is valid for all time without checking for unrealistic behaviour; examine the solution for large t and state any restrictions.