Evaluation of Definite Integrals by Substitution

Introduction

The construction of the definite integral definition begins with a continuous function f(x) on the closed interval [a, b]. The provided interval is divided into “n” subintervals that can be assumed of equal length (Δx), which is not required. 

In each subinterval, an arbitrary domain value, x i is picked, and its subsequent function value, f(x i), is calculated. Each function value is multiplied by the length of the relevant subinterval, and the sum of these “n” products is calculated which is known as Riemann sum. 

A Riemann sum is a total that can be positive, negative, or zero, based on how the function acts on a closed interval. 

If f(x) > 0 on [a, b], for example, the Riemann sum is a positive real number. 

If f(x) < 0 on [a, b], the Riemann sum is a negative real number.

The limit of Riemann sum when the n tends to infinity is called the integral.

Integration by substitution is a common integration approach employed when the function to be integrated is either a complex function or when direct integration is not possible. This integration method by substitution simplifies the integral of a function by converting the provided function to a simpler function.

Integration by Substitution

Integration by substitution is utilised when the integration of a given function cannot be accomplished directly because the algebraic function is not in the standard form. Furthermore, the provided function can be reduced to its standard form by substituting appropriately. 

Consider the indefinite integral,∫f(x).dx, of a function f(x) for evaluation. By replacing x with g(t) and substituting, this integral can be changed into another form, as follows:

x=g(t).

I = ∫f(x).dx

x = g(t) where dx/dt = g'(t)

dx = g'(t).dt

I =∫f(x).dx=∫f(g(t)).g′(t).dt

Performing Substitutional Indefinite Integration

Integration by substitution, or substitutional integration, may be performed by following these steps:

• Step 1: For the given function to be decreased, choose a new variable, t.
• Step 2: Evaluation of the value of dx, where f(x) is integrated with respect to x, for the provided integral.
• Step 3: Substitute the new value dx for the needed substitution in the function f(x).
• Step 4: Integrate the function derived after substituting.
• Step 5: To get the final solution, replace the initial variable x with the final answer.

Important Substitutions Used in Substitutional Integration

The following are some of the most useful substitutes for simplifying the provided statement and making the integration process easier.

x = aSinθ or x = aCosθ is used for the integral function f(a2−x2)

x = a Secθ or x = aCosecθ is used for the integral of the function f(x2−a2)

x = aTanθ, or x = aCotθ is used for the integral of the function f(x2+a2). f(a2+x2) 

x = a Cos2θ is used for the integral of the functions f( a-xa+x ), f( a+xa-x )

Evaluation of Definite Integrals by Substitution

The steps for evaluation of ab f(x) dx via substitution are as follows:

• Step 1: To reduce the provided integral to a known form, consider the integral without limitations and insert y = f (x) or x = g(y).
• Step 2: Without stating the integration constant, integrate the new integrand with regard to the new variable.
• Step 3: Substitute the new variable for the old one and write the result in terms of the old one.
• Step 4: Find the values of the answers obtained in (3) at the stated integral limits, as well as the difference between them.
• Step 5: Find the range’s top and lower limits.

Example

Using the substitution approach, evaluate the integral using the given substitution, 

(a) ʃ01 [x/(x2 + 1)] dx

Solution:

(a) Let x2 + 1 = t

=> 2x dx = dt

=> x dx = dt/2

If x = 0, t = 1 and when x = 1, t = 2

Hence, ʃ01 [x/(x2 + 1)] dx = (1/2)ʃ12 dt/t

                                     = (1/2)[log t]21

                                     = (1/2)[log 2 – log 1]

                                     = (log 2)/2

Conclusion

The antiderivative and indefinite integrals of a function are closely connected to the definite integral. The indefinite integral, if it exists, has a real numerical value, whereas the latter two represent an unlimited number of functions that vary only by a constant.

Definite integration by Substitution is one of the most efficient and commonly used ways of solving complex integration problem. It reduces the problem to a known integral that can be easily solved with changed limits.