Euler's Substitutions JEE Notes | EduRev

Mathematics (Maths) Class 12

JEE : Euler's Substitutions JEE Notes | EduRev

The document Euler's Substitutions JEE Notes | EduRev is a part of the JEE Course Mathematics (Maths) Class 12.
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Euler's Substitutions

Integrals of the form   Euler`s Substitutions JEE Notes | EduRevdx are calculated with the aid of the three Euler substitutions.
Euler`s Substitutions JEE Notes | EduRev

Ex.82 Evaluate I = Euler`s Substitutions JEE Notes | EduRev


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 JEE Notes | EduRev

Euler`s Substitutions JEE Notes | EduRev

Euler`s Substitutions JEE Notes | EduRev

Euler`s Substitutions JEE Notes | EduRev

Euler`s Substitutions JEE Notes | EduRev

Euler`s Substitutions JEE Notes | EduRev

Ex.83 Evaluate Euler`s Substitutions JEE Notes | EduRev

Sol.

Euler`s Substitutions JEE Notes | EduRev ...(1)

Euler`s Substitutions JEE Notes | EduRev ...(2)

We know 

Euler`s Substitutions JEE Notes | EduRev

Euler`s Substitutions JEE Notes | EduRev

Euler`s Substitutions JEE Notes | EduRev

Euler`s Substitutions JEE Notes | EduRev

Euler`s Substitutions JEE Notes | EduRev

Euler`s Substitutions JEE Notes | EduRev

Euler`s Substitutions JEE Notes | EduRev

I. CAN WE INTEGRATE ALL CONTINUOUS FUNCTION ?
The questions arises : Will our strategy for integration enable us to find the integral of every continuous function ? For example, can we use it to evaluate Euler`s Substitutions JEE Notes | EduRev ? 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 books 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, substraction multiplication, division, and composition for instance, the function f(x) = Euler`s Substitutions JEE Notes | EduRev is an elementary function
If f is an elementary function, then f' is an elementary function but Euler`s Substitutions JEE Notes | EduRevf(x) dx need not be an elementary function. Consider f(x) = Euler`s Substitutions JEE Notes | EduRev Since f is continuous, its integral exists, and if we define the function F by Euler`s Substitutions JEE Notes | EduRev  then we know from part 1 of the fundamental theorem of calculus that F'(x) = Euler`s Substitutions JEE Notes | EduRev Thus, f(x) = Euler`s Substitutions JEE Notes | EduRev 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 JEE Notes | EduRevEuler`s Substitutions JEE Notes | EduRev dx in term of the function we know. The same can be said of the following integrals.
Euler`s Substitutions JEE Notes | EduRev

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