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CIE A-Level Maths Study Notes

1.8.1 Fundamental Concept of Integration

Integration, a cornerstone of calculus, is essentially the reverse of differentiation. It's a process used for finding the original function when its derivative is known. This section delves into the integration of functions in the form (ax+b)n(ax+b)^n, where nn is a rational number, excluding -1. This covers the integration of constant multiples, sums, and differences of functions.

Understanding Integration

Integration as the Reverse of Differentiation:

This concept involves determining a function when its derivative is known, effectively reversing the process of differentiation.

Notation and Terminology:

  • The integral sign, "∫", represents the operation of integration.
  • The expression to be integrated, followed by "dx", indicates integration with respect to x.
  • "dx" in integration corresponds to the "dx" in the derivative notation dydx\frac{dy}{dx}.

Types of Integrals

  • Indefinite Integrals: These are integrals without specific limits, including a constant of integration, +c+c, to account for the loss of the original function's vertical position during differentiation.
  • Definite Integrals: Integrals with specified upper and lower limits, used for calculating exact values, such as areas under curves. They exclude the constant +c+c.

Integration Formulas

  • Basic Integral Form:
axndx=axn+1n+1+c,where n1.\int a x^n \, dx = \frac{a x^{n+1}}{n+1} + c, \quad \text{where } n \neq -1.
  • Integral of a Function in the Form (ax+b)n(ax+b)^n:
axndx=axn+1n+1+c\int a x^n \, dx = \frac{a x^{n+1}}{n+1} + c

again, n1.n \neq -1.

Worked Examples

Example 1:

Find the equation of a curve yy in terms of xx that passes through the point (1, 3), givendydx=6x2\frac{dy}{dx} = 6x^2.

Solution:

1. Integration:

y=6x2dx=6x33+c=2x3+c.y = \int 6x^2 \, dx = \frac{6x^{3}}{3} + c = 2x^3 + c.

2. Determining Constant cc:

Using the point (1, 3):

3=2(1)3+c.3 = 2(1)^3 + c.

Solving for cc:

c=1.c = 1.

3. Final Equation:

$y = 2x^3 + 1. <p></p><h3><strong>Example2:</strong></h3><p>Calculatetheintegralof<p></p><h3><strong>Example 2:</strong> </h3><p>Calculate the integral of 5x^2 - 2x + 3.</p><p></p><p><strong>Solution:</strong></p><p><strong>1.IntegralCalculation</strong>:</p>.</p><p></p><p><strong>Solution:</strong></p><p><strong>1. Integral Calculation</strong>: </p>\int (5x^2 - 2x + 3) \, dx = \frac{5x^3}{3} - \frac{2x^2}{2} + 3x + c = \frac{5x^3}{3} - x^2 + 3x + c.<p></p><p><strong>2.SimplifiedEquation:</strong></p><p></p><p><strong>2. Simplified Equation: </strong></p>\therefore \int (5x^2 - 2x + 3) \, dx = \frac{5x^3}{3} - x^2 + 3x + c.$


Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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