Integration By Tables
TLDRThis video tutorial demonstrates the method of integration by tables, a technique used to find the indefinite integral of complex functions. It walks through three example problems, showing how to identify and apply the appropriate formula for each case, and emphasizes the importance of converting the integral expression from x to u variables. The video also highlights the need to adjust for differing values of du/dx when substituting variables and concludes with the final answers for each integral.
Takeaways
- 📚 The video discusses finding the indefinite integral using integration by tables, a method for solving complex integrals.
- 🔍 The first example is the integral of 1/(x^2) * √(1-x^2), which requires identifying the correct formula from calculus resources.
- 🎯 The formula used for the first example is ∫(1/u^2) * √(a^2 - u^2) du = -1/(a^2u) * √(a^2 - u^2) + C, where u = x and a = 1.
- 🌟 The process involves substituting x with u and dx with du, and then applying the formula to find the integral.
- 📈 The second example is the integral of du/(u^2 * √(u^2 + a^2)), with u^2 = 4x^2 and a^2 = 9, leading to u = 2x and du = 2dx.
- 🔗 In this case, the integral is transformed to match the formula, with adjustments for the u and a values, and dx is replaced with du/2.
- 📝 The final result for the second example is -√(9 + 4x^2)/ (9x) + C, after substituting the values of u and a into the formula.
- 🌐 The third example is the integral of x^2/√(5 - 4x^2), which also requires finding the appropriate formula from a calculus textbook or online resources.
- 🔑 The formula used for the third example is ∫u^2/(√(a^2 - u^2)) du = -u/(2√(a^2 - u^2)) + (a^2/2) * arcsin(u/a) + C, with u = 2x and a = √5.
- 🧩 The integral is again transformed to match the formula, with the necessary substitutions and adjustments for the u and a values.
- 🏆 The final result for the third example is (1/8) * (x * √(5 - 4x^2) - 5 * arcsin(2x/√5)) + C, after substituting and simplifying the expression.
Q & A
What is the main topic of the video?
-The main topic of the video is how to find the indefinite integral using integration by tables.
What is the first example problem presented in the video?
-The first example problem is finding the integral of 1 divided by x squared, times the square root of 1 minus x squared, dx.
What formula is needed to solve the first example problem?
-The formula needed for the first example is the integral of 1 over u squared, times the square root of a squared minus u squared, du, which is equal to negative 1 over a squared u, times the square root of a squared minus u squared, plus c.
How is the expression converted to contain the u variable in the first example?
-In the first example, x squared and u squared are the same, so u is equal to x and du is equal to dx.
What is the second example problem worked on in the video?
-The second example problem is finding the indefinite integral of d u divided by u squared, times the square root of u squared plus a squared.
How are the values of u and a identified in the second example?
-In the second example, u squared is 4x squared, so u is the square root of 4x squared which is 2x, and a squared is 9, so a is the square root of 9 which is 3.
What is the significance of the formula used in the second example?
-The formula used in the second example helps to identify the values of u and a, and allows for the conversion of the integral expression from x variable to the u variable for application of the formula.
What is the third example problem discussed in the video?
-The third example problem is finding the indefinite integral of x squared divided by the square root of 5 minus 4x squared.
How is the expression changed to contain the u variable in the third example?
-In the third example, a 4 is placed in front of x squared to turn it into u squared, and dx is replaced with du divided by 2.
What is the final result of the third example problem?
-The final result of the third example is 1/8 times the negative of the square root of 5 minus 4x squared, plus 16 times the arc sine of 2x divided by the square root of 5, plus c.
What is the key takeaway from the video regarding integration by tables?
-The key takeaway is to identify the right formula and then convert the integral expression from the x variable to the u variable before applying the formula to find the indefinite integral.
Outlines
📚 Introduction to Integration by Tables
This paragraph introduces the concept of finding the indefinite integral using the method of integration by tables. It begins with an example problem, which is finding the integral of 1 divided by x squared, times the square root of 1 minus x squared, with respect to x (dx). The speaker explains the need to identify the appropriate formula, which can be found in a calculus textbook or online resources. The formula needed for this example is the integral of 1 over u squared, times the square root of a squared minus u squared, du. The explanation continues with the process of substituting x squared for u squared, and dx for du, to solve the integral. The final answer is given as negative square root of 1 minus x squared over x, plus a constant of integration, c.
🔍 Identifying Formulas and Substitution
In this paragraph, the focus is on identifying the correct formula for integration by tables and the process of substitution. The speaker presents another example, the integral of du divided by u squared, times the square root of u squared plus a squared. The speaker emphasizes the importance of identifying the values of u and a, and how to convert the integral expression from x variable to u variable. The example provided involves u squared as 4x squared and a squared as 9, leading to u as 2x and du as 2dx. The speaker then demonstrates how to apply the formula and solve for the integral, resulting in the final answer of negative square root of 9 plus 4x squared, divided by 9x, plus a constant of integration, c.
🧮 Solving Complex Integrals with u-Substitution
The paragraph discusses solving a more complex integral involving x squared divided by the square root of 5 minus 4x squared. The speaker guides through the process of finding the appropriate formula from the textbook, which is the integral of u squared divided by the square root of a squared minus u squared du. The speaker then explains how to identify u as 2x and a as the square root of five, and how to convert the integral expression to contain the u variable. The solution involves multiplying the top and bottom by 4 to get rid of the x squared and turn it into u squared. The final answer is derived by substituting the values of u and a into the formula, resulting in an expression involving the square root of five minus four x squared, and an arc sine term, plus a constant of integration, c.
Mindmap
Keywords
💡Indefinite Integral
💡Integration by Tables
💡Formula
💡Variable Substitution
💡u and du
💡Constant 'a'
💡Square Root
💡Differential 'dx'
💡Arcsine Function
💡Constant of Integration
Highlights
The video discusses finding the indefinite integral using integration by tables, a method for solving complex integrals.
The integral of 1/(x^2) * sqrt(1 - x^2) is used as an example to demonstrate the process.
The appropriate formula for the example is identified as integral(1/u^2) * sqrt(a^2 - u^2) du = -1/(a^2u) * sqrt(a^2 - u^2) + C.
The expression is converted into one containing the u variable, where x^2 = u^2 and u = x, d u = dx.
The final answer for the first example is -sqrt(1 - x^2)/x + C, showcasing the application of the formula.
The second example involves finding the integral of du/(u^2 * sqrt(u^2 + a^2)) with u^2 = 4x^2 and a^2 = 9.
For the second example, d u is not equal to dx, so the expression must be converted to incorporate 2x dx.
The integral is manipulated to match the formula by multiplying the top and bottom by four and replacing dx with du/2.
The final answer for the second example is -sqrt(9 + 4x^2)/ (9x) + C, demonstrating the method's effectiveness.
The third example involves the integral of x^2/sqrt(5 - 4x^2), using a different formula for integration by tables.
In the third example, a^2 is found to be 5, u is 2x, and du is 2dx, requiring a conversion of the expression.
The integral is transformed by multiplying the top and bottom by 4 and replacing dx with du/2.
The final answer for the third example is 1/8 * (x * sqrt(5 - 4x^2) - 5 * (2x/sqrt(5)) + C), illustrating the versatility of the method.
Integration by tables is a powerful technique for solving integrals that do not fit standard formulas.
The method requires identifying the correct formula, substituting variables, and applying the formula accurately.
The video emphasizes the importance of recognizing when d u does not equal dx and adjusting the expression accordingly.
The examples provided in the video demonstrate the step-by-step process of integration by tables, making it easier to understand and apply.
The video is a valuable resource for those learning calculus and seeking to understand the integration by tables method.
Transcripts
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