Higher Maths: Integration, Areas and Differential Equations

Integration at Higher goes well beyond integrating powers of $x$. This material covers the integrating functions, definite integrals and applying integral calculus skills listed in the Higher Mathematics course specification, and it assumes you can already integrate simple polynomials.

The first group of questions extends the techniques. You integrate brackets of the form $(px+q{)}^n$, including negative and fractional powers such as $\frac{1}{(3x-1{)}^3}$ and $\sqrt{4x+1}$, where the classic error is to forget to divide by the coefficient $p$. You integrate $p\cos (qx+r)$ and $p\sin (qx+r)$, keeping track of the minus sign that comes with the integral of sine and of the division by $q$. Every result can be checked by differentiating, and the explanations show that check.

Definite integrals then use limits that are integers, radians, surds and fractions, as the course specification lists. Several are marked non-calculator, as in question paper 1, and rely on exact values such as $\sin \frac{\pi }{2}=1$ and $\cos \frac{\pi }{3}=\frac{1}{2}$.

The area questions find the area between a straight line and a curve and between two curves: you locate the intersection points to get the limits, decide which graph is on top, and integrate the upper function minus the lower one. A further question shows why an area that lies partly below the $x$-axis must be split at the root: a single integral over the whole interval can give 0 even though the area is not zero.

Finally, you solve differential equations of the form $\frac{dy}{dx}=f(x)$, using a point on the curve to find the constant of integration, and determine a function from a rate of change and an initial condition, such as the height of a growing plant.

The material offers three formats. The quiz has 12 questions whose explanations show each step and name the slip behind the typical wrong options. The flashcards collect the standard integrals, the rule for areas between graphs, and the method for differential equations. The printable written work has 8 longer tasks to set out by hand, including an area between two curves with surd limits, a trigonometric definite integral with an exact answer, a curve found from its gradient function and a written explanation of why an integral and an area can differ.

It is aimed at S5 and S6 students preparing for Higher Mathematics, after the category's mixed-practice material, which covers the area under a simple curve.

• Integrate functions of the form (px + q)ⁿ, n ≠ −1
• Integrate p sin(qx + r) and p cos(qx + r)
• Evaluate definite integrals with integer, radian, surd and fractional limits
• Find the area between a straight line and a curve, and between two curves
• Handle areas that lie partly below the x-axis
• Solve differential equations of the form dy/dx = f(x) with an initial condition
• Determine a function from a given rate of change

Practice material written by Zestly, based on the Qualifications Scotland (formerly SQA) Higher Mathematics course specification, version 3.0 (calculus skills: integrating functions; using integration to calculate definite integrals; applying integral calculus).