Lecture 30: Integration by Parts

1.1 Integration by Parts

1.1.1 Basic Concepts

#####Definition: Distributedintegrationmethod

description: let $u(x),\nu(x)$ have continuous first-order derivatives, then $\int udv=uv-\int vdu$

Explanation

• Where: sometimes $udv$ simple, sometimes $vdu$ simple
• The selection of v and u is the core point;

1.1.2 Example Problems

Example Question: $\int xe^xdx$

• Analysis
• Since the integral of $ex$ is relatively easy to find, you need to consider how to place ex in the vdu position;
• But you can also consider placing x in the VDU position, though it's a bit more difficult;
• Analysis
• $\int xe^xdx=\int x\operatorname{de}^x=xe^x-\int e^x\operatorname{dx}$
• $\int\mathrm{e}^{\mathrm{x}}\mathrm{d}\mathrm{x}=\times\mathrm{e}^{\mathrm{x}}-\mathrm{e}^{\mathrm{x}}+\mathrm{d}$
• Question Type: PartialIntegralMethod

Example Question: $\int x sin xdx$

• Analysis
• Analysis
• The original form equals: $-\int xdcosx=-\left[x\cos x-\int\cos dx\right]$
• Question Type: Distributedintegrationmethod

Example Question: $\int sec^3 xdx$

• Analysis

-

• Analysis
• Original form: $\int\sec^{3}xdx=\int\sec xd\tan x=\sec xdx-\int\tan^{2}x\sin xdx$ $=\sec x\tan x-\int\sec^3xdx+\int\sec xdx$
• At this point, since a negative sec 3 xdx also appears on the right side of the original expression, it can be moved to the right
• $\int sec^3x\mathrm{~d}x=\frac12\left[secxtanx+\ln|\sec x+\ln x|\right]+\mathrm{C}$
• Part 1 and 2 are because the sec on the right has been moved to the left;
• Question Type: PartialIntegralMethod

1.2 Summary of Integration by Parts

Summary

• Concept: Reverses the multiplication formula process from the previous differentiation to form the distributed integration method;

When to use

• Suitable formultiplying two different functions;
• 1. $\int xe^{x}dx$
• 2. $\int x\sin xdx$
• 3. $\int e^{x}\sin xdx$

How to Use

• Eight types of typical integration rules for parts;
• Multinomial function × Exponent|Trigonometry:
• 1. $\int p_n(x)e^{ax}\operatorname{d}x$
• Score the index
• 2. $\int p_n(x)\sin ax\operatorname{d}x$
• Triangle Together
• 3. $\int p_n(x)\cos axdx$
• Triangle Together
• Multinomial Functions × Logarithm | Inverse Triangle:
• 4. $\int P_n(x)\ln xdx$
• Fill in polynomials
• 5. $\int P_n(x)\arctan xdx$
• Fill in polynomials
• 6. $\int P_n(x)\arcsin xdx$
• Fill in polynomials
• Exponential × Trigonometric Function
• 7. $\int e^{\alpha x}\sin\beta xdx$
• Either factor can be chosen first when applying integration by parts.
• 8. $\int e^{\alpha x}\cos\beta xdx$
• Either factor can be chosen first when applying integration by parts.