Summary Feynman Technique in Integration – Advanced Problem-Solving Handout by Satyabrat Sahu

Understanding Feynman Integration

Matthew Blaine

September 9, 2023

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,Contents

1 Introduction to Feynman Integration and the Leibniz Rule 9
1.1 History of Feynman Integration . . . . . . . . . . . . . . . . . . . 9
1.2 How it Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Derivation of the Leibniz Rule . . . . . . . . . . . . . . . . . . . . 10
1.4 Summary and Next Steps . . . . . . . . . . . . . . . . . . . . . . 11
R1 2
2 Evaluation of 0 xln−1x dx 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Defining a New Function . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Evaluating the Function at Key Points . . . . . . . . . . . . . . . 13
2.4 Differentiating under the Integral Sign . . . . . . . . . . . . . . . 14
2.5 Integrating to Obtain the Original Function . . . . . . . . . . . . 14
2.6 Determining the Constant of Integration . . . . . . . . . . . . . . 14
2.7 Solution of the Integral . . . . . . . . . . . . . . . . . . . . . . . . 14
2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
R ∞ −2x −8x
3 Evaluation of 0 e −e x dx 15
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Defining a New Function . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 Evaluating the Function at Key Points . . . . . . . . . . . . . . . 15
3.4 Differentiating under the Integral Sign . . . . . . . . . . . . . . . 16
3.5 Integrating to Obtain the Original Function . . . . . . . . . . . . 16
3.6 Determining the Constant of Integration . . . . . . . . . . . . . . 16
3.7 Solution of the Integral . . . . . . . . . . . . . . . . . . . . . . . . 16
3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
R∞
4 Evaluation of 0 sin(ax)
x dx 19
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 Defining a New Function . . . . . . . . . . . . . . . . . . . . . . . 19
4.3 Evaluating the Function at Key Points . . . . . . . . . . . . . . . 19
4.4 Differentiating under the Integral Sign . . . . . . . . . . . . . . . 20
4.5 Integrating to Obtain the Original Function . . . . . . . . . . . . 20
4.6 Determining the Constant of Integration . . . . . . . . . . . . . . 20
4.7 Solution of the Integral . . . . . . . . . . . . . . . . . . . . . . . . 20

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4.8
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
R∞ 2
5 Evaluation of 0 sinx2 x dx 23
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.2 Defining a New Function . . . . . . . . . . . . . . . . . . . . . . . 23
5.3 Evaluating the Function at Key Points . . . . . . . . . . . . . . . 23
5.4 Differentiating under the Integral Sign . . . . . . . . . . . . . . . 24
5.5 Integrating to Obtain the Original Function . . . . . . . . . . . . 24
5.6 Determining the Constant of Integration . . . . . . . . . . . . . . 24
5.7 Solution of the Integral . . . . . . . . . . . . . . . . . . . . . . . . 25
5.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
R∞
6 Evaluation of 0 sinx x dx 27
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.2 Defining a New Function . . . . . . . . . . . . . . . . . . . . . . . 27
6.3 Evaluating the Function at Key Points . . . . . . . . . . . . . . . 28
6.4 Differentiating under the Integral Sign Using the Leibniz Rule . . 28
6.5 Integrating to Obtain the Original Function . . . . . . . . . . . . 28
6.6 Determining the Constant of Integration . . . . . . . . . . . . . . 28
6.7 Solution of the Integral . . . . . . . . . . . . . . . . . . . . . . . . 29
6.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

7 A new way to express factorials 31
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
7.2 Defining a function of t . . . . . . . . . . . . . . . . . . . . . . . 31
7.3 Taking multiple derivatives . . . . . . . . . . . . . . . . . . . . . 31
7.4 Evaluating the function at t=1 . . . . . . . . . . . . . . . . . . . 32
7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
R∞
8 Evaluation of 0 cos(ax)−cos(bx)
x2 dx 33
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
8.2 Defining a New Function . . . . . . . . . . . . . . . . . . . . . . . 33
8.3 Evaluating the Function at Key Points . . . . . . . . . . . . . . . 33
8.4 Differentiating under the Integral Sign . . . . . . . . . . . . . . . 34
8.5 Integrating to Obtain the Original Function . . . . . . . . . . . . 34
8.6 Determining the Constant of Integration . . . . . . . . . . . . . . 34
8.7 Solution of the Integral . . . . . . . . . . . . . . . . . . . . . . . . 34
8.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

9 Evaluating the Gaussian Integral 37
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
9.2 Defining a New Function . . . . . . . . . . . . . . . . . . . . . . . 37
9.3 Evaluating the Function at Key Points . . . . . . . . . . . . . . . 37
9.4 Differentiating under the Integral Sign . . . . . . . . . . . . . . . 38
9.5 Integrating from 0 to Infinity . . . . . . . . . . . . . . . . . . . . 38
9.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39