Integration by Substitution

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Differentiation from First Principles

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Rates of Change

Conceptual Understanding of Rates of Change A rate is a comparison between two quantities with different units. We often judge[...]

Applications of Maximum and Minimum

Example 1 Find the maximum value of $i=100\sin(50 \pi t +0.32)$, and the time when this maximum occurs. Related YouTube[...]

Solving Quadratic Equations by Factors

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When No Need to Apply Quotient Rule for Differentiating a Fraction

The following examples show that you do not need to apply the quotient rule for differentiating when the denominator is[...]

Perfect Numbers

Some of the first patterns of natural numbers spotted and studied by Pythagoreans were $\textit{perfect numbers}$. These are natural numbers[...]

Method of Exhaustion

Eudoxus, Greek mathematician 408-355 BC, developed the idea of seeking mathematical solutions using he $\textit{Method on Exhaustion}$: at this stage,[...]

The Golden Ratio

Since the days of old, artists as well as mathematicians have known that there is a special, aesthetically pleasing rectangle[...]

Digital Cubes

The number $153$ is equal to the sum of the cubes of its digits: $$1^3 + 5^3 + 3^3 =[...]

Laws of Exponents (Index Laws)

$\textbf{Laws of Exponents (Index Laws)}$ $a^x \times a^y = a^{x+y}$ To $\textit{multiply}$ numbers with the $\textit{same base}$, keep the base[...]

Proof by Contradiction

$\textbf{Introduction to Proof by Contradiction}$ The basic idea of $\textit{Proof by Contradiction}$ is to assume that the statement that we[...]

The Sign of One

Make the following numbers using FOUR 1s using any mathematics operators and/or symbols, such as $\dfrac{x}{y}$, $\sqrt{x}$, decimal dots, $+$,[...]

Number Curiosity

Are these numbers beautiful? \begin{align} \displaystyle 1 \times 8 + 1 &= 9 \\ 12 \times 8 + 2 &=[...]

Nicomachus Theorem

$\textbf{Nicomachus}$ discovered "Nicomachus Theorem" interesting number patterns involving cubes and sums of odd numbers. Nicomachus was born in Roman Syria[...]

How to get top marks in VCE Mathematical Methods Units 3 & 4

Pass your VCE Mathematical Methods Units 3 & 4 with flying colours You aspire to get high marks in your[...]

Tips on how to study VCE specialist mathematics Units 3 & 4

You don’t need to be a genius to pass VCE specialist mathematics units 3 & 4 I’m sure you’ve read[...]

How to improve on your VCE Mathematics

Improve your VCE Mathematics skills The Victorian Certificate of Education (VCE) in mathematics can be challenging, many students find themselves[...]

Follow these Preliminary Chemistry study tips and success will be guaranteed

Preliminary Chemistry Study Tips For finding Preliminary Chemistry Study Tips, preliminary chemistry marks for year eleven may not count towards[...]

Hack your way to success in HSC Chemistry

Success in HSC Chemistry We all want a good marks in chemistry, not just to make you feel good but[...]

How to Ace the HSC Maths Exams

Tips for Success in HSC Maths, Extension 1, Extension 2, and Maths General Our team works with students all over[...]

Preliminary Physics Study Guide for year 11

Preliminary Physics Study Guide The preliminary physics syllabus is required study for your 11 students in New South Wales Australia[...]

How to Succeed in IB Maths

Your One-Stop, Five-Step Guide for SL and HL Maths Victory There’s no feeling like the pride of completing a course[...]

How to succeed at HSC Physics Exam

The HSC physics Exam is a requirement for year 12 students in New South Wales. This university entrance exam otherwise[...]

Logarithmic Differentiation

Basic Rule of Logarithmic Differentiation $$ \displaystyle \dfrac{d}{dx}\log_e{x} = \dfrac{1}{x} \\ \dfrac{d}{dx}\log_e{f(x)} = \dfrac{f'(x)}{f(x)} $$ Practice Questions Question 1 Differentiate[...]

Binomial Expansion | Binomial Theorem

Binomial Expansion is based on two terms, that is binomial. Any expression of the form \( (a+b)^n \) is called[...]

Compound Interest | Series and Sequences

Compound Interest is being used to calculate the total investment across over time. Suppose John invests $1000 in the bank.[...]

Geometric Sequence | Math Help

A geometric sequence is also referred as a geometric progression. Each term of a geometric sequence can be obtained from[...]

Arithmetic Sequence | Maths Help

An Arithmetic Sequence is a sequence in which each term differs from the previous one by the same fixed number.[...]

General Term of a Number Sequence

The general term of a number sequence is one of many ways of defining sequences. Consider the tower of bricks.[...]

Understanding Number Sequence

Number Sequence or progression is ordered list of numbers defined by a pattern or rule. The numbers in the sequence[...]

Implicit Differentiation | Calculus Help

Implicit Differentiation for Calculus Problems This very powerful differentiation process follows from the chain rule. $$u = g(f(x)) \\ \frac{du}{dx}[...]

Surd Equations Reducible to Quadratic | Math Algebra

Surd Equations Reducible to Quadratic | Math Algebra Surd Equations Reducible to Quadratic for Math Algebra is done squaring both[...]

Trigonometric Equations Reducible to Quadratic | Math Skills

Trigonometric Equations Reducible to Quadratic for Math Skills Trigonometric Equations Reducible to Quadratic for Math Skills are based on trigonometric[...]

Exponential Equations Reducible to Quadratic | Math Help

Exponential Equations Reducible to Quadratic for Math Help Exponential Equations Reducible to Quadratic for Math Help is based on various[...]

Fraction Equations Reducible to Quadratic | Free Math Help

Fraction Equations Reducible to Quadratic | Free Math Help iitutor provides full explains of Fraction Equations Reducible to Quadratic for[...]

Algebra Equations Reducible to Quadratic | Math Help

Algebra Equations Reducible to Quadratic Form | Math Help Algebra Equations Reducible to Quadratic Form for Math Help is done[...]

Best Examples of Mathematical Induction Inequality

Mathematical Induction Inequality Proofs Mathematical Induction Inequality is being used for proving inequalities. It is quite often applied for the[...]

Best Examples of Mathematical Induction Divisibility

Mathematical Induction Divisibility Proofs Mathematical Induction Divisibility can be used to prove divisibility, such as divisible by 3, 5 etc.[...]

5 Important Patterns of First Principles

First Principles The First Principles defines how basic rule of derivative of a function. $$\displaystyle f'(x) = \lim_{h\to\infty} \frac{f(x+h)-f(x)}{h}$$ Patterns[...]

10 Deadly Common Algebra Mistakes

Common Patterns of Algebra Mistakes Students often make Common Algebra Mistakes due to confusions such as expand and simplify rules,[...]

Mathematical Induction Fundamentals

Mathematical Induction Fundamentals The Mathematical Induction Fundamentals are defined for applying 3 steps, such as step 1 for showing its[...]

Integrating Trigonometric Functions by Double Angle Formula

Integrating Trigonometric Functions by Double Angle Formula Integrating Trigonometric Functions can be done by Double Angle Formula reducing the power[...]

Trigonometric Integration by Substitution

Trigonometric Integration by Substitution Trigonometric Integration by Substitution can be handled using the basic rules such as \( \displaystyle \frac{d}{dx}[...]

Definite Integration by Substitution

Definite Integration by Substitution Definite Integration by Substitution requires to convert upper and lower limits of definite integration. $$ \displaystyle[...]

Integration by Substitution

Integration by Substitution Integration by Substitution is performed by replacing the pronumeral (variable) to other pronumeral to simplify the expression[...]

Definite Integrals

Background of Definite Integrals Definite Integrals are calculated by subtracting the function values of upper and lower limits. $$\displaystyle \int_{a}^{b}{f(x)}dx[...]

Indefinite Integral of Rational Functions

Understanding Indefinite Integral of Rational Functions Using Indefinite Integral of Rational Functions requires that the format of the expression must[...]

Radical Indefinite Integrals

Radical Indefinite Integrals should be performed after converting its radical or surd notations into index form. $$\displaystyle \sqrt[n]{x^m} = x^{\frac{m}{n}}$$[...]

Indefinite Integral Formula

Basics of Indefinite Integral Formula Indefinite Integral Formula ahs been made from the reverse operations of differentiation or anti-differentiation. \([...]

Geometric Sequence using Logarithms

Applications of Geometric Sequence using Logarithms Geometric Sequence using Logarithms is being used for finding the number of terms of[...]

4 important types of absolute value equation

Absolute Value Equation There are 4 main type of absolute value equation regarding whether there are; absolute value and a[...]

Logarithmic Inequalities

Solving logarithmic inequalities, it is important to understand the direction of the inequality changes if the base of the logarithms[...]

Quartic Graph Sketching

Quartic Graph Sketching is based on their concavities and x-intercepts to determine the basic shape of the quartic graphs. Question[...]

Cubic Graph Sketching

Basics of Cubic Graph Sketching Cubic Graph Sketching start considering from the \(x\)-intercepts and whether the leading coefficient is either[...]

Harder Factorise

Algebra - Harder Factorise or Factoring can be done by collecting common factors and using sums and products of factors.[...]

Sketching Quadratic Graphs

Sketching Quadratic Graphs using Transfomration Sketching Quadratic Graphs are drawn based on \( y=x^2 \) graph for transforming and translating.[...]

Internal Division

Internal Division of Line Segments Internal Division an interval in a given ratio is that if the interval \( (x_1,y_1)[...]

Angle Between Lines

Linear Functions - Angle Between Lines If the acute angle \( \theta \) between two straight lines \( y =[...]

Solving Radical Equations

Solving radical equations are required to isolate the radicals or surds to one side of the equations. Then square both[...]

Integration Reverse Chain Rule

By recalling the chain rule, Integration Reverse Chain Rule comes from the usual chain rule of differentiation. This skill is[...]

Inequality with Variables in the Denominator

Solving Inequality with Variables in the Denominator requires special cares due to the direction of the inequalities. Let's have a[...]

12 Patterns of Logarithmic Equations

Solving logarithmic equations is done many ways using properties of logarithmic functions, such as multiply of logs, change the base[...]

Geometric Series for Time Payments

Time payments is are calculated based on Geometric Series for reducible compound interests. Basically geometric series formula is used for[...]

Complex Numbers with Vector Addition

Complex Numbers with Vector Addition is obtained using the sides of triangles, the sum of sides of two sides of[...]

Rates of Change involving Integration

Rates of Change for finding height and time can be solved by integration. Once integrated, find the integral constant using[...]

Three Dimensional Trigonometry

One of the most effective ways to solve Three Dimensional Trigonometry questions is to list all of the trigonometric ratios[...]

Quadratic Equations in Square Roots

For solving Quadratic Equations in Square Roots, it is required to square both sides entirely, but not individually. For instance,[...]

Area of a Triangle using Radius and Perimeter

Area of a triangle can be calculated using its perimeter and the radius of the circle which is inscribed in[...]

Collision of Projectile Motion

Collision of Projectile Motion is handled by setting the same vertical height and horizontal distance at a given time. Worked[...]

Collinear Proof in Circle Geometry

Collinear Proof can be done in Circle Geometry by showing Three or more points lie on a single straight line.[...]

Integration by Parts

Integration by Parts is made of product rule of differentiation. The derivative of \(uv\) is \(u'v + uv'\) and integrate[...]

Volumes by Cross Sections

Volumes by Cross Sections can be setting a unit volume, \( \delta V \) of a cross section. Then establish[...]

Mathematical Induction Inequality Proof with Factorials

Mathematical Induction Inequality Proof with Factorials uses one of the properties of factorials, \( n! = n(n-1)! = n(n-1)(n-2)! \).[...]

Cyclic Quadrilateral in Circle Geometry

Cyclic Quadrilateral is inscribed into a circle, whose vertices all lie on a circle. The properties of Cyclic Quadrilateral in[...]

Locus of Complex Numbers

Locus of Complex Numbers is obtained by letting \( z = x+yi \) and simplifying the expressions. Operations of modulus,[...]

Mechanics Circular Motions

Mechanics Circular Motions are handled by resolving forces horizontally and vertically in conjunction with the tension of the string, normal[...]

Ellipse Geometry

Ellipse Geometry is used for proving ellipse questions relating to geometry such as similar triangles and circle geometry. Worked Examples[...]

Inequality Proofs

Inequality Proofs can be done many ways. For proving \(A \ge B \), one of the easiest ways is to[...]

Complex Number Regions

Regions defined by complex numbers \( \displaystyle z = x + yi \) where \(x\) and \(y\) are real numbers,[...]

Ellipse Discriminant Eccentricity

Discriminant can be used to ellipses for identifying the status of their intersections in conjunction with eccentricity. The eccentricity of[...]

Rationalising Denominators

Rationalising Denominators simplifies the fractions by multiplying the conjugates of the denominators. Worked Example of Rationalising Denominators Simplify \( \displaystyle\frac{1}{\sqrt{1}[...]

Probability with Replacement

Probability with Replacement is used for questions where the outcomes is returned back to the sample space again. Which means[...]

Integration Recurrence Formula

For Integration Recurrence Formula or reduction formula, it is important set a relationship between two consecutive terms by using mostly[...]

Polynomial Equation and Maximum Coefficient

If \( \alpha \) is a root of a polynomial equation \( P(x) = 0 \), then \( P(\alpha) =[...]

Absolute Value Inequalities

Absolute Value Inequalities are usually proved by the absolute value of a certain value is greater than or equal to[...]

Integration using Trigonometric Properties

Trigonometric properties such as the sum of squares of sine and cosine with the same angle is one, $$ \displaystyle[...]

Circle Geometry with Semicircles

There are many properties of circle geometry with semi circles, such as equal arcs on circles of equal radii subtend[...]

Volumes by Cylindrical Shells Method

Let’s consider the problem of finding the volume of the solid obtained by rotating about the \(x\)-axis or parallel to[...]

Integrating Binomial Expansion

Integrating Binomial Expansion is being used for evaluating certain series or expansions by substituting particular values after integrating binomial expansion.[...]

Probability Ratio using Combination

Counting Techniques for Probability Ratio using Combination The probability ratio of an event is the likelihood of the chance that[...]

Inequalities using Arithmetic Mean Geometric Mean

Arithmetic Mean of \(a\) and \(b\) is always greater than or equal to the Geometric Mean of \(a\) and \(b\),[...]

Mathematical Induction Inequality Proof with Two Initials

Usually, mathematical induction inequality proof requires one initial value, but in some cases, two initials are to be required, such[...]

Trigonometric Proof using Compound Angle Formula

There are many areas to apply the compound angle formulas, and trigonometric proof using compound angle formula is one of[...]

Finding a Function from Differential Equation

The solution of a differential equation is to find an expression without \(\displaystyle \frac{d}{dx} \) notations using given conditions. Note[...]

3 Ways of Evaluating Nested Square Roots

Nested square roots or nested radical problems are quite interesting to solve. The key skill for this question is to[...]