Back-substitution is a method used to solve systems of equations, particularly in the context of solving integrals through substitution. After performing a substitution to simplify the integral, back-substitution involves replacing the variable in the resulting expression with its original expression to find the solution in terms of the original variable. This process is essential in ensuring that the final answer aligns with the variables and limits of the original problem.
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Back-substitution is crucial when dealing with definite integrals, as it allows for proper evaluation within specified limits.
The substitution step may alter variables, but back-substitution reverts these changes, ensuring clarity in the solution.
It's important to correctly identify and use the inverse of the substitution function during back-substitution.
Back-substitution helps to simplify complex integrals into manageable forms, making it easier to evaluate them.
Failure to properly apply back-substitution can lead to incorrect results, making it essential to double-check this step.
Review Questions
How does back-substitution aid in evaluating definite integrals after a substitution has been made?
Back-substitution helps in evaluating definite integrals by allowing you to revert to the original variable after simplifying the integral. Once a substitution has been performed, back-substitution replaces the simplified variable with its original expression. This step is necessary because it ensures that you can accurately evaluate the integral within its specified limits, leading to a correct final answer.
What are common mistakes made during back-substitution, and how can they be avoided?
Common mistakes during back-substitution include failing to correctly revert to the original variable or miscalculating the inverse of the substitution function. These errors can be avoided by carefully tracking your substitutions and double-checking each step of your work. Always ensure that you replace every instance of the new variable with its corresponding expression from the substitution step, keeping in mind any limits if dealing with definite integrals.
Evaluate a specific integral using substitution and explain each step including back-substitution.
To evaluate an integral such as $$\int (2x) e^{x^2} dx$$, start by substituting $$u = x^2$$, which leads to $$du = 2x \, dx$$. The integral transforms into $$\int e^u \, du$$. After evaluating, you obtain $$e^u + C$$. Now, applying back-substitution means replacing $$u$$ with $$x^2$$ again, resulting in $$e^{x^2} + C$$. This illustrates how back-substitution brings us back to our original variable for clarity and correctness in our final answer.