A Level ExamSolutions Maths Edexcel

This subject is broken down into 227 topics in 3 modules:

  1. Pure 159 topics
  2. Statistics 27 topics
  3. Mechanics 41 topics
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This page was last modified on 28 September 2024.

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ExamSolutions Maths

Pure

Algebra and Functions – Rational Expressions: Simplifying – Simplifying algebraic fractions

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Algebra and Functions – Rational Expressions: Simplifying – Simplifying algebraic fractions

Algebra and Functions – Rational Expressions: Simplifying – Simplifying Algebraic Fractions

Key Concepts

  • Algebraic fractions, also known as rational expressions, are fractions in which the numerator and the denominator are both polynomials.
  • These fractions can often be simplified by factoring and cancelling common factors in the numerator and the denominator.
  • The process of simplifying an algebraic fraction is similar to that of simplifying a numeric fraction: you divide the numerator and the denominator by their highest common factor.

Factoring

  • The first step in simplifying an algebraic fraction is typically to factor both the numerator and the denominator.
  • Factoring breaks down the expressions into the product of their factors, which can simplify the process of identifying and cancelling common factors.
  • Remember that a difference of squares, such as a^2 - b^2, can be factored into (a - b)(a + b).

Cancelling Common Factors

  • Once the numerator and denominator have been factored, the next step is to identify and cancel any common factors.
  • Common factors are expressions that appear in both the numerator and the denominator. By definition, any expression divided by itself is 1, so common factors can be cancelled out.
  • This step is crucial: cancelling common factors can greatly simplify the fraction and highlight its important features.

Multiplying and Dividing Algebraic Fractions

  • When multiplying algebraic fractions, you simply multiply the numerators together and the denominators together. Simplify if possible.
  • When dividing algebraic fractions, you multiply by the reciprocal of the divisor. Realise that this is the same as multiplying by the fraction upside-down. Simplify if possible.

Adding and Subtracting Algebraic Fractions

  • Like numeric fractions, algebraic fractions can only be added or subtracted if they have the same denominator (i.e., they are like fractions).
  • If the fractions have different denominators, you will need to find a common denominator before you can add or subtract them.

Useful Strategies

  • Practice makes perfect. Spend time working on a variety of problems involving algebraic fractions to solidify your understanding.
  • Watch out for complex fractions - fractions where the numerator and/or denominator itself contains fractions. You'll need to simplify these before you can proceed.
  • Stick with it. Simplifying algebraic fractions can be a complex process, but it's also a foundational skill in much of algebra. Mastering this will set you up for success in more complex topics.

Course material for ExamSolutions Maths, module Pure, topic Algebra and Functions – Rational Expressions: Simplifying – Simplifying algebraic fractions

ExamSolutions Maths

Pure

Integrals of sin x, cos x, sec² x

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Integrals of sin x, cos x, sec² x

Understanding the Integrals

  • Integration involves finding the anti-derivative of a function, meaning finding the function that when differentiated, gives the original function.
  • Integral of sin x is -cos x + C where C represents constant of integration.
  • The integral of cos x is sin x + C.
  • The integral of sec² x is tan x + C.
  • Remember that the operation of integration reverses the operation of differentiation.

Calculating the Integrals

  • To calculate the integral of sin x, antidifferentiate sin x to -cos x and add a constant of integration, C.
  • Similarly, to calculate the integral of cos x, antidifferentiate cos x to sin x and add the constant of integration, C.
  • To find the integral of sec² x, antidifferentiate sec² x to tan x and again add the constant of integration, C.
  • The constant of integration accounts for any vertical shift that may occur during antidifferentiation.

Example Problem

  • Given an example integral of sin x dx.
  • Antidifferentiate sin x to -cos x.
  • Don't forget to add the constant of integration, so the solution to the integral of sin x dx is -cos x + C.

Trigonometric Substitutions

  • Besides standard integrals, be ready to tackle integrals involving products or powers of sine and cosine. This might involve trigonometric substitution or integration by parts.
  • Always consider the power rules for sin x and cos x, if the exponent is negative or fractional.

Real World Applications

  • Understanding integrals of sin x, cos x, sec² x is applicable in physics, engineering, and computer science where these integrals are used to solve real world problems.
  • The integral of cos x is used in signal analysis and the integral of sin x is used in calculating power series.

Study Tips

  • Consistently practice solving these kinds of integrations. As a routine, aim to calculate the integral without referring to the base formula.
  • Developing a good understanding of trigonometric identities and their derivatives will significantly ease the process of computing these integrals.
  • Utilise a variety of resources such as textbooks, online lecture notes, or instructional videos to understand and solve complex integrations. Ensure you understand the underlying principles of integration to succeed in your problem-solving process.

Course material for ExamSolutions Maths, module Pure, topic Integrals of sin x, cos x, sec² x

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