U-substitution With Definite Integrals
The Organic Chemistry Tutor・2 minutes read
To evaluate definite integrals using u substitution, start by integrating the function and making u equal to x squared plus 4, which allows you to replace x variables with u variables. Adjust the lower and upper limits using the u values corresponding to the x values given, and then evaluate the antiderivative of u squared from 4 to 8 to find the final answer, along with other examples that illustrate the process.
Insights
- By substituting u for x squared plus 4 in definite integrals, the process involves replacing x variables with u variables, adjusting limits accordingly, and evaluating the antiderivative to obtain the final answer.
- Utilizing u substitution in definite integrals with varying u expressions such as 16 minus x squared or 1 plus x squared requires finding du, replacing x variables with u variables, and integrating within the specified limits to derive the final result.
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Recent questions
How do you evaluate definite integrals using u substitution?
By integrating the function with u substitution, replace x variables with u variables and adjust limits accordingly.
What is the process of setting u equal to a function in definite integrals?
Setting u equal to a function allows for replacing x variables with u variables to simplify integration.
How do you find the antiderivative of u squared in u substitution?
To find the antiderivative of u squared, integrate the function with respect to u and evaluate the limits accordingly.
What is the significance of finding du in u substitution?
Finding du allows for replacing x variables with u variables and simplifying the integration process in definite integrals.
How do you simplify the final answer in u substitution for definite integrals?
Simplify the expression obtained after integration to find the final answer of the definite integral using u substitution.
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Summary
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Evaluating Definite Integrals Using u Substitution
- To evaluate definite integrals using u substitution, start by integrating the function and making u equal to x squared plus 4, which allows you to replace x variables with u variables.
- Adjust the lower and upper limits using the u values corresponding to the x values given, and then evaluate the antiderivative of u squared from 4 to 8 to find the final answer of 448 over 3.
- In another example, set u equal to 16 minus x squared, find du as -2x dx, and replace x variables with u variables to solve for dx and integrate the expression from 16 to 0, resulting in the final answer of 256 over 3.
- For a third example, make u equal to 1 plus x squared, find du as 2x dx, and replace x variables with u variables to evaluate the definite integral from 2 to 5, leading to the final answer of 21 over 200 after simplifying the expression.
