Euler's Substitutions

Definition

Integrals of the form   dx are calculated with the aid of the three Euler substitutions.

Ex.1 Evaluate I = 

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 : 

Ex.2 Evaluate 

Sol.

 ...(1)

 ...(2)

We know 

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  ? 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 (a^x), 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) =  is an elementary function
If f is an elementary function, then f' is an elementary function but f(x) dx need not be an elementary function. Consider f(x) =  Since f is continuous, its integral exists, and if we define the function F by   then we know from part 1 of the fundamental theorem of calculus that F'(x) =  Thus, f(x) =  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   dx in term of the function we know. The same can be said of the following integrals.