A Simple Note on the Substitution in Definite Integrals

Differentials and integrals form the very basis of mathematical studies. This method in itself poses us with a relative advantage. It helps us in solving the integrals of functions regardless of the limits that the integral has on it, or whether the integral is definite or indefinite. This method specifically involves altering the limits of an integral for our convenience, so it is not sensitive to limit change. And, that makes this theorem even more effective compared to any other method of solving integrals. 

Substitution

Substitution is a method that is used to solve the integrals for a function. Using this method for solving indefinite integrals gives us a generalised equation as our answer. Using that answer we can easily compute an integral that is similar compared to what we solved previously. 

This method is also applicable for solving the definite integrals. However, to solve definite integrals, the limits for the integral of the function must be introduced in the given problem. When solving the definite integral using this method, the limits of the integration also change similar to the change in the function that is to be integrated. 

This method for calculating the integrals of functions becomes even more effective to use when the function in question carries a function along with its derivative in it. In that case, a certain valve is used to interchange with the function. And then a suitable process is carried out to find the solution to that question. 

As this method comprises a change in variables for integration, this method is also known as the u-substitution method, or reverse chain-rule method. That is because this method is quite similar to how we evaluate chain rule backwards. 

Substitution method

Let g(x) = u, and f(x) be a continuous function over the range of ‘g’. F(x) is the antiderivative of the function f(x). Also, g'(x) happens to be continuous over an interval.

 Based on these characteristics, the substitution rule for integration is given as:

∫ f [g(x)] g'(x) dx = ∫ f(u) du

Also,

∫ f(u) du = F(u) + C

where ‘C’ is the constant of integration,

And, F(u) stands for F [ g(x) ]

This method can also be specified in the following terms;

Let f, F, g, and u be the terms that are specified in the theorem above. 

Taking the derivative of F [ g(x) ] with respect to ‘x’

d / dx { F [ g(x) ] } = F’ [ g(x) ] g'(x) = f [ g(x) ] g'(x)

Now, integrating both the sides of the above equation with respect to ‘x’:

∫ f [ g(x) ] g'(x) dx = F [ g(x) ] + C

Now, substituting g(x) by ‘u’, and g'(x) dx by ‘du’:

∫ f [g(x)] g'(x) dx = ∫ f(u) du = F(u) + C = F [ g(x) ] + C

This happens to be the result of the substitution rule for integration. And it can be seen that this rule does not superimpose any specific limits.

Substitution of definite integrals

The result of the substitution rule in integration is given as follows:

∫ f [g(x)] g'(x) dx = ∫ f(u) du = F(u) + C = F [ g(x) ] + C

Here, f(x) stands for a function,

F(x) is the antiderivative of that function,

dx is the value to which it is to be integrated

‘C’ is the arbitrary constant or the constant of integration

‘x’ stands for a variable in the given function

Using the substitution method evaluate :01 xe4x² + 3

Now, assuming 4x² + 3 to be equal to ‘u’,

And, 8x dx to be equal to ‘du’.

Now adjusting the limits of the integration, using the given upper and lower limits of the integral;

When x = 0, u = 3

also, when x = 1, u= 7

Using the substitution method for integration, we get, 

01 xe4x² + 3 dx = 1 / 8 37eu du 

=> 1 / 8 [eu ]37

=> (e7 – e3) / 8

=> 134.568

The substitution method is specifically used to find the integration of a function even if the original value of that function is not known to us. And therefore, it breaks the chain rule that is posed over an integral. That is why this method is also known as the anti-chain rule.

Conclusion

The substitution method for integrals is used to solve the integration problems at which the normal conventional methods for properties of integration fail. This method can solve the integrals in which the integral limit is a differential variable. And, because of certain advantages that are posed over the integrals because of the action of this theorem, it is preferred over any other form. Also, this theorem can be used to solve problems related to indefinite integrals as well as definite integrals.