Question

How to know when to use integration by substitution vs. integration by parts?

Answer 1

Hint:
Here, we will show the difference between the two methods. Then with the help of examples we will be able to explain which method is used in which type of question. Integration is defined as the summation of all the discrete data. Integration is the inverse of differentiation and hence it is called antiderivative.

Complete step by step solution:
An integral can be solved by either using the substitution method or by parts. Now, in order to know which method is to be used in which question, we should observe whether if we do the substitution, we will be able to find the derivative or not. If we are able to do so, then, the substitution method is the one which should be followed.
But, if after doing the substitution, the question becomes more complex. Then, we should go for by parts keeping in mind that if we are given a product of two entirely different types of functions then we have to use the ILATE rule (Inverse, log, algebraic, trigonometric and exponential)
For instance, let us consider two examples.
Case 1: When we are given that $I = \int {{e^x} \cdot \cos xdx} $
Here, we can clearly observe two different types of functions and no matter whichever technique we apply, we will not be able to use the substitution method and find the derivative.
Hence, these types of questions should be solved using integration by parts.
Case 2: When we are given that $I = \int {{e^{\sin x}} \cdot \cos xdx} $
Now, if we observe we can find two trigonometric functions. Thus, there exists a possibility of substitution.
Now, we usually try to reduce the power. Thus, we will let $\sin x = u$
Now, differentiating both sides with respect to $x$, we get,
$\cos xdx = du$
Hence, substituting this value in the question, we get,
\[I = \int {{e^u}du} \]
Thus, this question just requires one more step to be solved.

Hence, we should know how to modify a question and by observation we can find out in which question, we are required to use which method.
Hence, this is the required answer.

Note:
In calculus, integration by substitution, also known as u-substitution or change of variables, is a method which is used for evaluating integrals or anti-derivatives. It is the counterpart to the chain rule for differentiation, in fact, it can also be considered as doing the chain rule "backwards". Also, integration by parts is a process of finding the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative.