Integration

Pure Mathematics 2 (9709) Study Notes: Chapter 2.5 Integration

Hello future Mathematician!

Welcome to the Integration chapter for Paper 2! If differentiation felt like driving a car forward, integration is like putting the car in reverse—you're working backward from the result (the gradient function) to find the original function.

In Pure 1, you learned the basics. Here in Pure 2, we expand those rules to handle more complex functions, especially those involving the exponential function, logarithms, and new trigonometric forms. We also learn a crucial numerical method—the Trapezium Rule—for estimating areas.

Don't worry if this seems tricky at first! We will break down the essential rules and methods step-by-step, focusing only on what the Paper 2 syllabus requires.

1. Integration as Reverse Differentiation (P2 Extensions)

The core principle of integration remains: it is the reverse process of differentiation. The key difference in P2 is applying this principle to functions where the inner part is a linear function, specifically in the form \((ax + b)\).

Key Concept: The "Divide by a" Rule

When you differentiate a function like \(f(ax+b)\), you multiply by the derivative of the bracket (\(a\)). Therefore, when you integrate \(f'(ax+b)\), you must divide by \(a\).

Common Mistake Alert: Always remember the Constant of Integration, \(+ C\), for indefinite integrals. If you forget it, you lose marks!

P2 Extended Integration Rules (Where \(n \neq -1\))

These are the mandatory extensions for Paper 2:

1. Power Rule Extension: (Used for polynomial terms)
\(\int (ax + b)^n dx = \frac{1}{a} \cdot \frac{(ax + b)^{n+1}}{n+1} + C\)

2. Exponential Functions:
\(\int e^{ax+b} dx = \frac{1}{a} e^{ax+b} + C\)
Example: \(\int 4e^{3x-1} dx = 4 \cdot \frac{1}{3} e^{3x-1} + C = \frac{4}{3} e^{3x-1} + C\)

3. Reciprocal Functions (Leads to \(\ln\)):
Differentiating \(\ln|x|\) gives \(\frac{1}{x}\), so integrating the reciprocal function gives \(\ln|x|\).
\(\int \frac{1}{ax+b} dx = \frac{1}{a} \ln|ax+b| + C\)
(Remember the modulus sign, \(| \dots |\), because you can only take the logarithm of a positive number!)
Example: \(\int \frac{5}{2x-7} dx = 5 \cdot \frac{1}{2} \ln|2x-7| + C = \frac{5}{2} \ln|2x-7| + C\)

4. Trigonometric Functions:
Recall: Differentiating \(\sin\) gives \(\cos\); differentiating \(\cos\) gives \(-\sin\).
\(\int \cos(ax+b) dx = \frac{1}{a} \sin(ax+b) + C\)
\(\int \sin(ax+b) dx = -\frac{1}{a} \cos(ax+b) + C\)
\(\int \sec^2(ax+b) dx = \frac{1}{a} \tan(ax+b) + C\)

Key Takeaway from Section 1:

P2 integration revolves around reversing the chain rule for linear functions. Integrate normally, then always divide by the coefficient of \(x\) from the inner function.

2. Integrating Functions Using Trigonometric Identities

Sometimes you cannot integrate a function directly, but you can if you rewrite it first. In Pure 2, this is mandatory for squared sine and cosine terms (like \(\sin^2 x\)) using double angle formulas.

Why do we need identities?

You know how to integrate \(\cos x\) or \(\cos(2x)\). But there is no simple rule for \(\int \cos^2 x dx\) or \(\int \sin^2 x dx\). We need to transform these squared terms into terms we can integrate.

Essential Identities for P2 Integration

You must use the double angle identities, specifically the rearranged forms:

1. For \(\cos^2 \theta\): Use \(\cos 2\theta = 2\cos^2 \theta - 1\).
Rearranged: \(\mathbf{\cos^2 \theta = \frac{1}{2}(1 + \cos 2\theta)}\)

2. For \(\sin^2 \theta\): Use \(\cos 2\theta = 1 - 2\sin^2 \theta\).
Rearranged: \(\mathbf{\sin^2 \theta = \frac{1}{2}(1 - \cos 2\theta)}\)

Step-by-Step Example

Example: Evaluate \(\int \sin^2(2x) dx\).

Step 1: Identify the angle. The angle here is \(2x\). The double angle identity uses \(2\theta\). If \(\theta = 2x\), then \(2\theta = 4x\).
Step 2: Apply the identity.
\(\sin^2(2x) = \frac{1}{2}(1 - \cos (2 \cdot 2x)) = \frac{1}{2}(1 - \cos 4x)\)
Step 3: Integrate the transformed function.
\(\int \frac{1}{2}(1 - \cos 4x) dx = \frac{1}{2} \int 1 dx - \frac{1}{2} \int \cos 4x dx\)
\(= \frac{1}{2} x - \frac{1}{2} \left( \frac{1}{4} \sin 4x \right) + C\)
\(= \frac{1}{2} x - \frac{1}{8} \sin 4x + C\)

Key Takeaway from Section 2:

If you see \(\sin^2(\dots)\) or \(\cos^2(\dots)\) in an integral, immediately reach for the relevant double angle formula to rewrite the expression before attempting to integrate.

3. The Trapezium Rule (Numerical Integration)

A definite integral \(\int_a^b f(x) dx\) represents the exact area under the curve \(y = f(x)\) between \(x=a\) and \(x=b\). When the area is too complex to find exactly, or when we are only given numerical data points, we use numerical methods to estimate the area.

What is the Trapezium Rule?

Instead of using rectangles (like in Riemann sums), the trapezium rule approximates the area by dividing the region into vertical strips (trapezia). The more strips you use, the more accurate your estimate becomes.

The Trapezium Rule Formula

To estimate \(\int_a^b f(x) dx\) using \(n\) strips (which means \(n+1\) ordinates or y-values):
1. Calculate the strip width: \(h = \frac{b - a}{n}\)
2. Apply the formula:
\(A \approx \frac{h}{2} [(y_0 + y_n) + 2(y_1 + y_2 + \dots + y_{n-1})]\)

Memory Aid: This formula means: \(\frac{h}{2}\) times [ (First Y + Last Y) + 2(Sum of all other Ys) ]

Step-by-Step Calculation

1. Calculate h: Determine the width of the strips based on the interval \([a, b]\) and the number of strips \(n\).
2. Create a Table: List the \(x\) values (starting at \(a=x_0\) and ending at \(b=x_n\)) and calculate the corresponding \(y\) values (\(y = f(x)\)).
3. Substitute into the Formula: Carefully identify the "end points" (\(y_0, y_n\)) and the "middle points" (\(y_1, y_2, \dots\)).
4. Calculate: Perform the calculation and round the final answer to the required accuracy (usually 3 significant figures).

Determining Over-estimates or Under-estimates

The syllabus requires you to use a sketch graph to determine if the trapezium rule provides an estimate that is larger (over-estimate) or smaller (under-estimate) than the true value. This depends on the concavity of the curve.

A trapezium line segment connects two points on the curve.

• Over-estimate: If the curve is concave up (like a smile, curving upwards from the tangent), the straight line of the trapezium lies above the curve.
  (Think: The shape holds water.)
• Under-estimate: If the curve is concave down (like a frown, curving downwards from the tangent), the straight line of the trapezium lies below the curve.
  (Think: The shape sheds water.)

Key Takeaway from Section 3:

The Trapezium Rule is a quick way to approximate area. The accuracy depends on the number of strips \(n\). You must be able to use the formula correctly and justify the nature of the error (over or under) using concavity and a simple sketch.