Integration practice questions (A-Level Maths)

Sample questions

The one free A-Level Maths question on this topic, from the free A-Level Maths Practice Paper. Each answer is behind its own toggle; the whole paper, its PDF and the full answer key are on the exam page.

1. Question 16 marks · Integration & area
   The curve with equation $y = 4x - x^2$ and the line with equation $y = x$ enclose a finite region $R$. (For a sketch: the curve is an inverted parabola through the origin with maximum at $(2, 4)$; the line passes through the origin with gradient 1 and cuts the curve again to the right, with the curve above the line between the two intersections.)
   $y = 4x - x^2, \qquad y = x$
   (a)

   Find the coordinates of the points of intersection of the curve and the line.

   [2]
   (b)

   Find the exact area of the region $R$.

   [4]
   Show answerFrom the free A-Level Maths Practice Paper →

Key terms

The definitions and formulas A-Level Maths expects for integration, typeset as they appear on the paper.

Indefinite integration

Reverse of differentiation: raise the power by one and divide by the new power, adding a constant. The constant of integration is a mark on its own.

$$\int x^n\,dx = \frac{x^{n+1}}{n+1} + c, \quad n \ne -1$$

Definite integrals and area

The integral between limits gives the signed area between the curve and the $x$-axis; regions below the axis come out negative, so split the integral at the roots and add the absolute values for total area.

$$\int_a^b f(x)\,dx = F(b) - F(a)$$

Area between a curve and a line

Integrate the difference between the upper and lower functions between their points of intersection, or subtract a triangle or trapezium found by geometry.

$$A = \int_a^b \big(f(x) - g(x)\big)\,dx$$

Standard integrals

The integrals of $e^{kx}$, $\frac{1}{x}$, $\sin kx$ and $\cos kx$. The modulus sign in $\ln|x|$ matters when the interval includes negative values.

$$\int e^{kx}\,dx = \frac{1}{k}e^{kx} + c, \quad \int \frac{1}{x}\,dx = \ln|x| + c, \quad \int \cos kx\,dx = \frac{1}{k}\sin kx + c$$

Integration by substitution

Replace a part of the integrand with $u$, change $dx$ to $du$ and the limits to values of $u$. Reverse chain rule results such as $\int f'(x)[f(x)]^n\,dx$ can be done by inspection.

$$\int f\big(g(x)\big)g'(x)\,dx = \int f(u)\,du$$

Integration by parts

For products such as $x e^x$, $x\sin x$ or $\ln x$ (take $u = \ln x$, $\frac{dv}{dx} = 1$). Choose $u$ as the factor that simplifies when differentiated.

$$\int u\frac{dv}{dx}\,dx = uv - \int v\frac{du}{dx}\,dx$$

Trapezium rule

Approximates a definite integral with $n$ strips of width $h$. Say whether it over- or under-estimates from the concavity of the curve.

$$\int_a^b y\,dx \approx \frac{h}{2}\Big[y_0 + 2(y_1 + \cdots + y_{n-1}) + y_n\Big], \qquad h = \frac{b - a}{n}$$

Separation of variables

For $\frac{dy}{dx} = f(x)g(y)$, collect the $y$ terms with $dy$ and the $x$ terms with $dx$, integrate both sides, and use the boundary condition to find the constant before giving $y$ explicitly if asked.

$$\int \frac{1}{g(y)}\,dy = \int f(x)\,dx$$

Frequently asked questions

Other A-Level Maths topics

• Proof, algebra and functions
• Coordinate geometry
• Sequences and series
• Trigonometry
• Exponentials and logarithms
• Differentiation
• Numerical methods and vectors
• Sampling and data presentation
• Probability and distributions
• Hypothesis testing
• Kinematics
• Forces, Newton's laws and moments

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