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Euler's Substitutions | Mathematics (Maths) Class 12 - JEE PDF Download

Definition

Integrals of the form   Euler`s Substitutions | Mathematics (Maths) Class 12 - JEEdx are calculated with the aid of the three Euler substitutions.
Euler`s Substitutions | Mathematics (Maths) Class 12 - JEE


Ex.1 Evaluate I = Euler`s Substitutions | Mathematics (Maths) Class 12 - JEE


Sol. In this case a < 0 and c < 0 therefore neither the first, nor the second, Euler substitution is applicable.
But the quadratic trinomial 7x – 10 – x2 has real roots α = 2, β = 5, therefore we use the third Euler substituion : 

Euler`s Substitutions | Mathematics (Maths) Class 12 - JEE

Euler`s Substitutions | Mathematics (Maths) Class 12 - JEE

Euler`s Substitutions | Mathematics (Maths) Class 12 - JEE

Euler`s Substitutions | Mathematics (Maths) Class 12 - JEE

Euler`s Substitutions | Mathematics (Maths) Class 12 - JEE

Euler`s Substitutions | Mathematics (Maths) Class 12 - JEE

Ex.2 Evaluate Euler`s Substitutions | Mathematics (Maths) Class 12 - JEE

Sol.

Euler`s Substitutions | Mathematics (Maths) Class 12 - JEE ...(1)

Euler`s Substitutions | Mathematics (Maths) Class 12 - JEE ...(2)

We know 

Euler`s Substitutions | Mathematics (Maths) Class 12 - JEE

Euler`s Substitutions | Mathematics (Maths) Class 12 - JEE

Euler`s Substitutions | Mathematics (Maths) Class 12 - JEE

Euler`s Substitutions | Mathematics (Maths) Class 12 - JEE

Euler`s Substitutions | Mathematics (Maths) Class 12 - JEE

Euler`s Substitutions | Mathematics (Maths) Class 12 - JEE

Euler`s Substitutions | Mathematics (Maths) Class 12 - JEE

Can We Integrate All Continuous Function?

The question arises: Will our strategy for integration enables us to find the integral of every continuous function? For example, can we use it to evaluate Euler`s Substitutions | Mathematics (Maths) Class 12 - JEE ? The answer is no, at least not in terms of the functions that we are familiar with.
The functions that we have been dealing with in this book are called elementary functions. These are the polynomials, rational functions, power functions (x3), exponential function (ax), logarithmic functions trigonometric and inverse trigonometric functions, hyperbolic and inverse hyperbolic functions, and all functions that can be obtained from these by the five operations of addition, subtraction multiplication, division, and composition for instance, the function f(x) = Euler`s Substitutions | Mathematics (Maths) Class 12 - JEE is an elementary function
If f is an elementary function, then f' is an elementary function but Euler`s Substitutions | Mathematics (Maths) Class 12 - JEEf(x) dx need not be an elementary function. Consider f(x) = Euler`s Substitutions | Mathematics (Maths) Class 12 - JEE Since f is continuous, its integral exists, and if we define the function F by Euler`s Substitutions | Mathematics (Maths) Class 12 - JEE  then we know from part 1 of the fundamental theorem of calculus that F'(x) = Euler`s Substitutions | Mathematics (Maths) Class 12 - JEE Thus, f(x) = Euler`s Substitutions | Mathematics (Maths) Class 12 - JEE has an antiderivative F, but it has been proved that F is not an elementary function.
?This means that no matter how hard we try, we will never succeed in evaluating  Euler`s Substitutions | Mathematics (Maths) Class 12 - JEEEuler`s Substitutions | Mathematics (Maths) Class 12 - JEE dx in term of the function we know. The same can be said of the following integrals.
Euler`s Substitutions | Mathematics (Maths) Class 12 - JEE
The document Euler's Substitutions | Mathematics (Maths) Class 12 - JEE is a part of the JEE Course Mathematics (Maths) Class 12.
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FAQs on Euler's Substitutions - Mathematics (Maths) Class 12 - JEE

1. What are Euler's substitutions?
Euler's substitutions are a set of mathematical techniques used to simplify and solve differential equations. These substitutions involve replacing variables or functions in a given differential equation with new variables or functions, which can make the equation easier to solve.
2. How do Euler's substitutions work?
Euler's substitutions work by transforming a given differential equation into a new form that is easier to solve. This is done by introducing new variables or functions and substituting them into the original equation. The new variables or functions are chosen in such a way that they simplify the equation or eliminate certain terms, making it more manageable to solve.
3. What are some common examples of Euler's substitutions?
Some common examples of Euler's substitutions include substituting the variable x with the new variable x = e^t, or substituting the function y with the new function y = vx, where v is a new variable. These substitutions can transform a given differential equation into a simpler form that can be easily solved using standard techniques.
4. Can Euler's substitutions be used for all types of differential equations?
No, Euler's substitutions may not be applicable to all types of differential equations. These substitutions are most commonly used for solving linear and homogeneous differential equations. They may not be effective for non-linear or non-homogeneous equations, where other techniques like variation of parameters or power series solutions may be more suitable.
5. Are there any limitations or drawbacks to using Euler's substitutions?
Yes, there are limitations to using Euler's substitutions. These substitutions may not always lead to a simplified equation or a solution that can be easily determined. In some cases, the substitutions may introduce additional complexities or make the equation more difficult to solve. It is important to carefully analyze the given differential equation and assess whether Euler's substitutions are the most appropriate technique for solving it.
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