Integration by Trigonometric Substitution

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Algebraic expressions Depending on the function we need to integrate, we can use this trigonometric expression as substitution to simplify the integral: 1. When $ displaystyle sqrt{{{{a}^{2}}-{{b}^{2}}{{x}^{2}}}}$ then substitute $ displaystyle x=frac{a}{b}sin theta $ and the helpful trigonometric identities is $ displaystyle {{sin }^{2}}x=1-{{cos }^{2}}x$ 2. When $ displaystyle sqrt{{{{a}^{2}}+{{b}^{2}}{{x}^{2}}}}$ then substitute $ displaystyle x=frac{a}{b}tan theta $ […]

Integration by Partial Fraction

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Integration by Partial Fraction If the function that need to be integrated is in the form of an algebraic fraction which is not easy to evaluate then you need to write the fraction into partial fraction to make it simpler for integration. You have to remember that this method is done only if the degree […]

Integration by Parts

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Integration by Parts Integration by parts method is used when we want to integrate the product of two functions. When finding the derivative of the product of two functions we use the product rule, and since the integral is the reverse of derivative then integration by parts is the reverse of the product rule. Let’s explain […]

Methods of Integration

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Methods of integrations Basic rules of integration or table of integration help us solve simple problem when the integral is given on standard form. But when the problem is more complicated we need more sophisticated methods like: Integration by Substitutions Integration by Parts Integration by Partial Fraction Decomposition Integration of using some Trigonometric Identities Integration of inverse […]

Integration by Substitution

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Integration by Substitution This method is also called the u-substitution or the reverse of chain rule of derivation. The chain rule except being useful in derivation is also in integration: If we have two functions $ displaystyle f(x)$ and $ displaystyle g(x)$ then the derivative of their composite function is:$ displaystyle (fcirc g{)}'(x)={f}'(g(x)){g}'(x)$. How it helps on integration […]

Integration Rules

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Integration rules We already know that finding an integral is the reverse of finding a Derivative. So firstly you should learn derivates We talked about two types of integral, but a more scientific definition is: Indefinite Integral $ displaystyle int{{f(x)dx=F(x)+C}}$ where $ displaystyle {F(x)}$ is an antiderivative of $ displaystyle {f(x)}$. What is an antiderivative? An antiderivative of […]

Introduction to Integration

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Introduction to Integration What is integration? The process of integration is the reverse of the process of Differentiation. It is labeled by the symbol $ displaystyle int{,}$ Integration is used to find areas, volumes and it helps in a lot of other things. One of the most used is in finding the area of an […]

Worked examples – Limits

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Worked Examples – Limits The indeterminate forms of limits $ displaystyle infty cdot infty ,infty cdot 0,frac{infty }{infty },frac{0}{0},frac{infty }{0},infty +infty ,infty -infty {{,1}^{infty }}{{,0}^{infty }},{{infty }^{0}}$ The indeterminate forms of limits $displaystyle infty cdot infty ,infty cdot 0,frac{infty }{infty },frac{0}{0},frac{infty }{0},$ $displaystyle infty +infty ,infty -infty ,{{1}^{infty }},{{0}^{infty }},{{infty }^{0}}$ Important Limits $ displaystyle […]

Examples of calculating the derivative

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Examples of calculating the derivative Example 1: Find the derivative of the functions.   Applying the constant rule:   If $ displaystyle y=c$ then $ displaystyle y’=0$ The Power function rule: If $ displaystyle y=a{{x}^{n}}$ then $ displaystyle y’=an{{x}^{{n-1}}}$ a) $ displaystyle y=3{{x}^{7}}$ $ displaystyle y’=(3cdot 7){{x}^{{7-1}}}=21{{x}^{6}}$ b) $ displaystyle y=-4$ $ displaystyle y’=(-4)’=0$ c) $ displaystyle […]

Derivative Rules

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Derivative Rules We know that if $displaystyle y=f(x)$ then the derivative is defined to be $displaystyle f'(x)=underset{{hto 0}}{mathop{{lim }}},frac{{f(x+h)-f(x)}}{h}$ Some notions we use when we write the derivative are: $ displaystyle y’=f'(x)=frac{{df}}{{dx}}=frac{{dy}}{{dx}}$ Determining the derivative of a function using the definition sometimes it  requires a lot of work and it’s easy to make mistakes so […]

The derivative using the limit definition

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The derivative using the limit definition Definition The derivative of the function $ displaystyle f$ at the point a is the limit when $ displaystyle hto 0$ of the function, if this limit exists. We label it f´(a) and $displaystyle f'(a)=underset{{hto 0}}{mathop{{lim }}},frac{{f(a+h)-f(a)}}{h}$ When the function$ displaystyle f$ is derivative on the point $ displaystyle […]

Combining Functions

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Combining Functions Combining Functions means performing basic arithmetic operations like addition, subtraction, multiplication and division with functions. Given two functions $ displaystyle f(x)$ and $ displaystyle g(x)$ we define: 1. The sum of two functions $displaystyle (f+g)(x)=f(x)+g(x)$ 2. The difference of two functions $ displaystyle (f-g)(x)=f(x)-g(x)$ 3. The product of two functions $ displaystyle (ftimes […]

Theorems on Limits

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Theorems on Limits It can be rather tedious to apply the $displaystyle varepsilon $ and $displaystyle delta $ limit test to individual functions. By remembering some basic theorems about limits we can avoid the some of this repetitive work. We shouldn’t forget that if a limit exists it is always unique.  “The Uniqueness of a Limit” […]

Introduction to Limits

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Introduction to Limits Numerical and Graphical approach to limits Numerical Approach Let’s take a function f(x) and see how the values of the functions change when x takes values closer to a specific number. Example: Let f(x)=3x+1 and calculate f(x) as x takes values closer to 1, but not exactly the value at 1. We first […]

The Limit of a Function

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The limit of a function Let f be a function and let c be a real number. We do not require that f be defined ar c but we do require that f be defined at least on a set of the form (c-p,c) U (c,c+p) with p>0). To say that $displaystyle underset{{xto c}}{mathop{{lim }}},f(x)=l$ is […]

The graph of a function

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The graph of a function We know that the graph of a function is the set of all points of the plan xOy that have like abscissa the faces (elements of the domain sets) and like ordinate they have the corresponding value of the function. The graph of the numeric function is the illustration of the […]

Function

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Function The relation f with first set A and second set B is called a function when every element of A is paired with only one element of B.  f:A→B We can write it based on the way we label the sets. You may find it written as f:X→Y If in the functional relation f with first set […]