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Substitution Rule for Indefinite Integrals Video Lecture | Calculus - Mathematics
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FAQs on Substitution Rule for Indefinite Integrals Video Lecture - Calculus - Mathematics
| 1. What is the substitution rule for indefinite integrals? |  |
Ans. The substitution rule, also known as the u-substitution method, is a technique used to evaluate indefinite integrals. It involves making a substitution by introducing a new variable, typically denoted as u, which allows us to simplify the integral and make it easier to solve.
| 2. How does the substitution rule work? | |
Ans. The substitution rule works by replacing the original variable in the integral with the new variable u. This substitution is chosen in such a way that it simplifies the integral and allows us to integrate it easily. After the substitution, we differentiate u with respect to the original variable and replace any remaining instances of the original variable with u. Finally, we integrate the resulting expression with respect to u and then substitute back the original variable if needed.
| 3. When should I use the substitution rule for indefinite integrals? | |
Ans. The substitution rule is particularly useful when dealing with integrals that involve compositions of functions, exponential functions, or trigonometric functions. It can also be used when the integrand contains a radical or a product of functions that can be simplified through substitution. If you encounter an integral that seems difficult to evaluate directly, trying the substitution rule is often a good strategy.
| 4. What are the steps involved in applying the substitution rule? | |
Ans. The steps for applying the substitution rule are as follows:
1. Identify a suitable substitution by selecting a new variable u.
2. Compute du/dx, the derivative of u with respect to the original variable x.
3. Replace all instances of the original variable x in the integral with the new variable u.
4. Replace dx with du in the integral.
5. Simplify the integral as much as possible using the substitution.
6. Integrate the resulting expression with respect to u.
7. Substitute back the original variable if necessary.
| 5. Can you provide an example of how to use the substitution rule for indefinite integrals? | |
Ans. Sure! Let's consider the integral ∫(2x + 1)² dx. To solve this integral, we can use the substitution rule. Let's make the substitution u = 2x + 1.
Differentiating u with respect to x gives du/dx = 2. Rearranging this equation, we have dx = du/2.
Substituting these values into the integral, we get ∫(2x + 1)² dx = ∫u² * (du/2).
Simplifying this expression, we have 1/2 ∫u² du.
Integrating u² with respect to u gives (1/2) * (u³/3) + C, where C is the constant of integration.
Finally, substituting back the original variable, we have (1/2) * ((2x + 1)³/3) + C as the solution to the original integral.
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Nov 22, 2024 Last updated |
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