Let “OPQ” be a triangle where angle POQ is , and it’s base be “x” and perpendicular “y” as shown in the picture below:
Pythagorian Identities
Then,
Appealing to the Pythagorian theorem, we have:
Now let us suppose op be “r” then:
And as:
and
We have:
Thus:
This is called the fundamental Pythagorian identity of trigon
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Properties of trigonometric functions:
It is often useful to remember and use the properties of trigonometric functions while applying trigonometry in real life.
The main properties among the properties of trigonometric functions are given below:
a> Quadrant rule of signs:
In first quadrant both abscissa and ordinate are positive or , , so sine , cosine , tangent and all other trigono
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Trigonometric Functions:
We have already defined the trigonometric functions such as sine , cosine , tangent of any magnitude already here:
trigonometric functions
Now , we consider trigonometric functions for angles of a right angled triangle , and see how they are a special case of trigonometric functions of angle of any magnitude.
Suppose ABC is a right angled triangle with hypotenuse &#
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Definite Integral:
Definite integral is a form of Integral or Anti derivative in which we don’t get a range of answer or indefinite answer , Instead we get a fixed or definite answer.
Or, A definite integral is the integral of a function in a closed interval and it is denoted by:
Which means , the Integral or Anti derivative of the function “f(x)” in an interval from “a
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Integration by Parts is a powerful tool and one of the best , most used techniques of integration used to evaluate integrals.
The main formula used in integration by parts is:
This formulas is derived from the product rule for differentiation as:
Multiplying both side of product rule formula by
or ,
Now integrating both side of above equation we get:
This formulas converts the problem of in
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integration by trigonometric substitution:
One of the most powerful techniques of integration is Integration by trigonometric substitution.
Integration by trigonometric substitution is similar technique to integration by substitution .
In integration by trigonometric substitution we substitute a variable by another trigonometrical variable.
We can integrate the integrals which involves , or .
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Integration by Substitution:
Integration can be easily done by using integration formulas , If the integral is in the standard form where we can easily apply formulas.
But if the function which is to be integrated is not in the standard form then it is either harder or impossible to use integration formulas to integrate.
In that case we need to to use Integration by Substitution method to int
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To find the antiderivative (Integral ) we use certain techniques and tricks , which
makes it easier to integrate a given function .
Such techniques used to find integral of a function effectively is called techniques of integration.
There are many techniques and tricks of integration.
Some the main techniques of integration are:
Techniques of integration:
1. By using Integration Formulas.
2. Integ
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Antiderivatives or Indefinite integrals:
If , “f” is a continuous function defined on an open interval (a,b) ;
Then the function “F” ( function F is capital “f”) is called antiderivative of function “f”, if the derivative of function “f” is function “F” on the interval.
Or , If ,
then , The function is said to be antider
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Exponential functions are the function which are defined in the form of:
f(x)=ax , where a is a constant and “x” is a variable.
The function “f(x) = ax“ is called an exponential function in base “a”.
The logarithmic functions are the inverse function of exponential function.
Or , if ” y = f(x) = ax ” then , x=f-1(y) is called the logarithmic functi
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