Integration by substitution, also known as u-substitution, is a method used to simplify and solve definite and indefinite integrals. It involves replacing a complicated expression with a new variable, making the integral easier to evaluate.
We use integration by substitution when we have an integral that involves a function within a function, or when we have an integral that involves a composite function. It is also useful when the integrand contains a product of functions.
To perform integration by substitution, we follow these steps:
1. Identify the inner function and let it equal a new variable, usually denoted by u.
2. Find the derivative of u with respect to x, du/dx.
3. Rewrite the integral in terms of u, replacing all instances of the inner function with u.
4. Substitute du/dx with du and dx with du/du.
5. Evaluate the integral in terms of u.
6. Finally, re-substitute u with the original inner function to obtain the solution in terms of x.
Integration by substitution allows us to solve integrals that would otherwise be difficult or impossible to evaluate. It also helps us to identify patterns and relationships between different functions, making it a useful tool for problem-solving in various fields such as physics, chemistry, and engineering.
Integration by substitution is not always applicable. It can only be used when the integral is in a specific form, involving a function within a function. In some cases, it may not be possible to find a suitable substitution, or the resulting integral may be just as difficult to evaluate. Additionally, it can be time-consuming and requires practice to become proficient at identifying the correct substitution.