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Math 141 Summer 2012 Exam 2 1. Use a trigonometric substitution to determine the infinite integral 10 pts. Each (a) Name: 6. X2 9 DX, x 3 x k1 1, (k + 1)(k + 2) ND a formula for the nth term of the
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How to fill out use a trigonometric substitution
How to fill out and use a trigonometric substitution:
01
Identify the integral you want to evaluate and determine if it can be simplified using a trigonometric substitution method.
02
Look for certain patterns or expressions within the integral that can be related to trigonometric identities.
03
If the integral contains terms in the form of √(a^2 - x^2), use a substitution of the form x = a sin(theta), where a is a constant and theta is a new variable.
04
If the integral contains terms in the form of √(a^2 + x^2), use a substitution of the form x = a tan(theta) or x = a sin(theta), where a is a constant and theta is a new variable.
05
If the integral contains terms in the form of √(x^2 - a^2), use a substitution of the form x = a sec(theta) or x = a tan(theta), where a is a constant and theta is a new variable.
06
Substitute the appropriate trigonometric expression into the integral and simplify the expression using trigonometric identities.
07
Evaluate the integral with the new trigonometric expression and solve for the original variable.
08
Sometimes, after integrating, you may need to convert the expression back into its original form by using inverse trigonometric functions.
09
Finally, simplify and verify your answer to ensure accuracy.
Who needs to use a trigonometric substitution:
01
Students studying advanced calculus or integration techniques often encounter trigonometric substitutions when trying to solve complex integrals.
02
Mathematicians and scientists who need to evaluate integrals that involve certain patterns or expressions that can be simplified using trigonometric substitutions.
03
Trigonometric substitutions can also be used in various engineering and physics applications when dealing with integrals in different coordinate systems or when solving differential equations.
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What is use a trigonometric substitution?
Trigonometric substitution is a technique used in integration to simplify integrals involving radicals and trigonometric functions.
Who is required to file use a trigonometric substitution?
Individuals studying calculus or advanced math topics may be required to use trigonometric substitution.
How to fill out use a trigonometric substitution?
To use trigonometric substitution, one must identify the appropriate trigonometric substitution, make the substitution, simplify the integral, and then back-substitute to get the final answer.
What is the purpose of use a trigonometric substitution?
The purpose of using trigonometric substitution is to make integrals involving radicals and trigonometric functions easier to solve.
What information must be reported on use a trigonometric substitution?
When using trigonometric substitution, one must report the substitution made, the steps taken to simplify the integral, and the final answer.
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