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Questions and Answers
What is the correct substitution used to evaluate the integral of $ rac{1}{1+e^x}$?
The primitive of $ae^x + be^{-x}$ is derived by which substitution?
After making the substitution $2 + 3logx = t$, what does the integral transform into?
In the evaluation of $ rac{1}{ ext{sqrt}(x+x)}$, which relationship is used for substitution?
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What advanced technique might be needed to evaluate the primitive of $ rac{2e^x - 3}{4e^x + 1}$?
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Study Notes
Evaluating Primitives with Substitution
- Substitution: A method to simplify integrals by replacing parts of the integrand with new variables.
- Goal of Substitution: To transform the integral into a simpler form that is easier to integrate.
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Steps Involved:
- Choose a Substitution: Select a part of the integrand to replace with a new variable.
- Differentiate: Differentiate the chosen substitution to find the differential of the new variable.
- Transform: Substitute the chosen variable and its differential into the integral, transforming it into a simpler form.
- Solve: Integrate the transformed integral.
- Substitute Back: Replace the new variable with its original expression to get the final answer.
Examples of Substitution
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Example 1: ∫(1/(1+e^x)) dx
- Substitution: e^x + 1 = t
- Differential: e^x dx = dt
- Transform: ∫(-dt/t)
- Solution: -log|t| + c
- Substitute Back: -log|e^x + 1| + c
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Example 2: ∫(ae^x + be^{-x}) dx
- Substitution: ae^x - be^{-x} = t
- Differential: (ae^x + be^{-x}) dx = dt
- Transform: ∫(dt/t)
- Solution: log|t| + c
- Substitute Back: log|ae^x - be^{-x}| + c
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Example 3: ∫(1/(2x + 3logx)) dx
- Substitution: 2 + 3logx = t
- Differential: (3/x) dx = dt, (dx/x) = dt/3
- Transform: ∫(dt/t)
- Solution: log|t| + c
- Substitute Back: log|2 + 3logx| + c
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Example 4:∫(1/√(x+x)) dx
- Substitution: 1 + √x = t
- Differential: (1/(2√x))dx = dt, dx/√x = 2dt
- Transform: ∫(2dt/t)
- Solution: 2log|t| + c
- Substitute Back: 2log|1 +√x| + c
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Example 5: ∫(2e^x-3)/(4e^x+1) dx
- Substitution: This example might require more advanced integration methods or techniques, such as trigonometric substitution.
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Description
Test your understanding of the substitution method for evaluating integrals. This quiz covers the steps involved in substitution, from choosing a variable to transforming and solving the integral. Practice with examples to solidify your knowledge of this essential calculus technique.
