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Questions and Answers
Which method in calculus is used for evaluating integrals and antiderivatives?
Which method in calculus is used for evaluating integrals and antiderivatives?
- Integration by substitution (correct)
- Differentiation
- Reverse chain rule
- Chain rule
What is the counterpart to the chain rule for differentiation?
What is the counterpart to the chain rule for differentiation?
- Integration by substitution (correct)
- Differentiation
- Change of variables
- Reverse chain rule
What is the result of the following indefinite integral: $\int (2x^3 + 1)^7 (x^2) dx$?
What is the result of the following indefinite integral: $\int (2x^3 + 1)^7 (x^2) dx$?
- $\frac{1},{6} (2x^3 + 1)^6 + C$
- $\frac{1},{9} (2x^3 + 1)^9 + C$
- $\frac{1},{10} (2x^3 + 1)^{10} + C$
- $\frac{1},{8} (2x^3 + 1)^8 + C$ (correct)
What is the value of $\frac{du},{dx}$ if $u = 2x^3 + 1$?
What is the value of $\frac{du},{dx}$ if $u = 2x^3 + 1$?
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What is the differential form of $du$ if $du = 6x^2 dx$?
What is the differential form of $du$ if $du = 6x^2 dx$?
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Study Notes
Integration and Antiderivatives
- Substitution method is used in calculus for evaluating integrals and antiderivatives.
Chain Rule Counterpart
- Substitution rule is the counterpart to the chain rule for differentiation.
Evaluating Indefinite Integrals
- The result of the indefinite integral ∫(2x³ + 1)⁷(x²)dx is achieved by using the substitution method.
Finding Derivatives
- If u = 2x³ + 1, then the value of du/dx is 6x².
Differential Form
- The differential form of du is 6x² dx, where du = 6x² dx.
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