Understand integration as the reverse process of differentiation and include an arbitrary constant in indefinite integrals

IGCSE Additional Mathematics (0606) – Comprehensive Lecture Notes

1. Quick‑review of prerequisite topics

1.1 Functions

• Definition & notation: \(y=f(x)\) gives the output for each admissible input \(x\).
• Domain & range: list all real numbers for which the expression is defined.
• One‑to‑one test (horizontal‑line test):

  • A function is one‑to‑one iff every horizontal line cuts its graph at most once.
  • If the test fails, restrict the domain so that the function becomes one‑to‑one (e.g. \(f(x)=\sqrt{x-2}\) has domain \(x\ge2\)).

• Inverse functions:

  • If \(f\) is one‑to‑one, the inverse is denoted \(f^{-1}\) and satisfies \(f^{-1}(f(x))=x\).
  • Graphically, the inverse is the reflection of the original graph in the line \(y=x\).
  • Example sketch: \(f(x)=\sqrt{x-2}\) and its inverse \(f^{-1}(x)=x^{2}+2\) (both drawn on the same axes).

• Composite functions:

  • Notation: \((f\circ g)(x)=f\bigl(g(x)\bigr)\).
  • When writing without parentheses, use a clear separator: \(f\,g\) is ambiguous, so prefer \(f\!\circ\!g\) or \(f(g(x))\).

• Typical graphs: linear, quadratic, exponential, logarithmic, basic trigonometric curves.

1.2 Quadratic functions

• Standard form: \(y=ax^{2}+bx+c\) with \(a\neq0\).
• Completing the square to obtain vertex form \(y=a(x-h)^{2}+k\).
• Discriminant \(\Delta=b^{2}-4ac\):

  • \(\Delta>0\) – two distinct real roots.
  • \(\Delta=0\) – one repeated real root (tangent to the \(x\)-axis).
  • \(\Delta<0\) – no real roots (parabola does not intersect the \(x\)-axis).

• Quadratic inequalities (required for the syllabus):

  • Solve \(ax^{2}+bx+c>0\) or \(<0\) by analysing the sign of the quadratic on intervals determined by its real roots.
  • Use a sign‑chart or the fact that a parabola opens upwards (\(a>0\)) or downwards (\(a<0\)).

• Maximum / minimum using differentiation:

  • If \(y=ax^{2}+bx+c\), then \(y' = 2ax+b\).
  • Set \(y'=0\) ⇒ \(x=-\dfrac{b}{2a}\). Substitute back to obtain the vertex \((h,k)\).
  • For \(a>0\) the vertex is a minimum; for \(a<0\) it is a maximum.

• Factor and remainder theorems – useful for checking roots and synthetic division.

1.3 Polynomials

• Degree, factor theorem, synthetic & long division, roots & multiplicities.

1.4 Logarithmic & exponential functions

• Definitions, laws of logarithms, change‑of‑base formula.
• Solving equations of the form \(a^{x}=b\) and \(\log_{a}b=c\).
• Key graphs: \(y=e^{x}\) and \(y=\ln x\).

1.5 Straight‑line graphs & circles

• Line equation \(y=mx+c\), gradient, intercepts, parallel/perpendicular conditions.
• Standard form of a circle \((x-h)^{2}+(y-k)^{2}=r^{2}\).

1.6 Circular measure & trigonometry

• Radian measure (the syllabus assumes radians for calculus).
• Six trig functions, key identities, exact values at \(0,\frac{\pi}{6},\frac{\pi}{4},\frac{\pi}{3},\frac{\pi}{2}\).

1.7 Vectors

• Notation \(\vec{u}= \langle u{1},u{2}\rangle\), magnitude \(|\vec{u}|=\sqrt{u{1}^{2}+u{2}^{2}}\).
• Addition, scalar multiplication, dot product \(\vec{u}\cdot\vec{v}=u{1}v{1}+u{2}v{2}\).

2. Differentiation – The forward process

For a function \(y=f(x)\) the derivative \(\dfrac{dy}{dx}=f'(x)\) gives the instantaneous rate of change.

3. Integration – The reverse process

3.1 Antiderivatives

If a function \(F(x)\) satisfies \(\displaystyle\frac{d}{dx}F(x)=f(x)\), then \(F\) is an antiderivative (or indefinite integral) of \(f\).

3.2 Notation

The whole family of antiderivatives of \(f(x)\) is written

\[

\int f(x)\,dx,

\]

where

• \(\int\) – integral sign
• \(f(x)\) – integrand
• \(dx\) – variable of integration

3.3 The constant of integration

Differentiation removes any constant term. When we “undo’’ differentiation we must add an arbitrary constant \(C\) to represent every possible antiderivative:

\[

\int f(x)\,dx = F(x)+C.

\]

The value of \(C\) is determined only when an extra condition (e.g. an initial value) is supplied.

3.4 Basic integration formulas (reverse of differentiation rules)

3.5 Integration techniques required by the syllabus

1. Substitution (reverse chain rule) – used when the integrand contains a function and its derivative.

   
   Example: \(\displaystyle\int 2x\,e^{x^{2}}dx\)

   
   Let \(u=x^{2}\Rightarrow du=2x\,dx\). Then \(\int e^{u}du=e^{u}+C=e^{x^{2}}+C\).

2. Integration of linear combinations – integrate each term separately and combine the constants.
3. Using initial conditions – determine the constant \(C\) after integration.

4. Worked examples (integration focus)

1. Example 1: \(\displaystyle\int 5x^{3}\,dx\)

   
   \(\displaystyle\int 5x^{3}\,dx = 5\cdot\frac{x^{4}}{4}+C = \frac{5}{4}x^{4}+C.\)

2. Example 2: \(\displaystyle\int \frac{2}{x}\,dx\)

   
   \(\displaystyle\int \frac{2}{x}\,dx = 2\ln|x|+C.\)

3. Example 3: \(\displaystyle\int (3e^{x}-4\cos x)\,dx\)

   
   \(\displaystyle\int 3e^{x}\,dx = 3e^{x}+C{1},\quad \int -4\cos x\,dx = -4\sin x + C{2}\)

   
   Collecting constants: \(\displaystyle\int (3e^{x}-4\cos x)\,dx = 3e^{x}-4\sin x + C.\)

4. Example 4 (initial condition): Find \(F(x)\) if \(F'(x)=2x\) and \(F(1)=5\).

   
   Integrate: \(F(x)=\int 2x\,dx = x^{2}+C.\)

   
   Apply the condition: \(1^{2}+C=5\Rightarrow C=4.\)

   
   Hence \(F(x)=x^{2}+4.\)

5. Example 5 (substitution): \(\displaystyle\int 2x\sin(x^{2})\,dx\)

   
   Let \(u=x^{2}\Rightarrow du=2x\,dx\). Then \(\displaystyle\int \sin u\,du = -\cos u + C = -\cos(x^{2})+C.\)

6. Example 6 (quadratic max/min via differentiation): Find the maximum value of \(y=-2x^{2}+8x-3\).

   
   Derivative: \(y'=-4x+8\). Set \(y'=0\Rightarrow x=2.\)

   
   Second derivative \(y''=-4<0\) ⇒ maximum.

   
   Maximum value: \(y(2)=-2(2)^{2}+8(2)-3 = -8+16-3 = 5.\)

5. Applications of integration

• Area under a curve (definite integral):

  \[

  \text{Area}= \int_{a}^{b} f(x)\,dx.

  \]

  For IGCSE the emphasis is on recognising that the antiderivative evaluated at the limits gives the required area.

• Motion problems:

  • If velocity \(v(t)=\dfrac{dx}{dt}\) is known, displacement is \(\displaystyle\int v(t)\,dt\).
  • Similarly, acceleration integrates to velocity.

• Finding the constant of integration from a given point, e.g. “the curve passes through \((2,3)\)”.

6. Common pitfalls

• Omitting the constant \(C\) in an indefinite integral.
• Applying the power rule when the exponent is \(-1\); the correct integral is \(\ln|x|+C\).
• Confusing \(\int f'(x)\,dx\) with \(f(x)\); the result is \(f(x)+C\).
• For substitution, forgetting to replace the differential correctly (the \(du\) must appear).
• Using degrees instead of radians when integrating trigonometric functions (the syllabus assumes radian measure).
• Assuming a function has an inverse without checking the one‑to‑one condition.

7. Summary tables

7.1 Differentiation ↔ Integration correspondence

7.2 Key integration formulas (IGCSE level)