Calculus Volume 2 · 免费预览
分部积分
分部积分是乘积法则的逆运算，它把乘积形式的被积函数拆分为 u 和 dv，使新的积分更为简洁；它还可用于对数函数、反三角函数，以及多项式与指数函数乘积的积分。
1. The formula and its origin
The formula and its origin
2. Logarithms and inverse functions
Logarithms and inverse functions
3. Repeated parts and the tabular method
Repeated parts and the tabular method
4. Definite integration by parts
Definite integration by parts
5. Formula origin
Integration by parts is the integral form of which differentiation rule?
6. LIATE priority
According to the LIATE mnemonic, which function should be chosen as u when integrating x times ln x?
7. Integral of ln x
Integration by parts on the integral of ln x dx chooses u = ln x and dv = dx. What is the result?
8. Cyclic case
Applying parts twice to the integral of e^x sin x dx returns the original integral I. Which equation results?
9. Tabular method purpose
What is the tabular method for repeated integration by parts?
10. Choosing dv
When integrating x times cos x by parts, which choice of dv is correct?
11. Parts pieces
In the parts formula integral of u dv equals uv minus integral of v du, how many pieces u and dv must be chosen initially? Answer with a number.
12. Integral of x cos x
Choosing u = x and dv = cos x dx gives du = dx and v = sin x. What is the integral of x cos x dx? Answer like x sin x + cos x + C.
13. Definite parts boundary
Definite integration by parts adds evaluation of uv at the limits. As two letters naming that term, what boundary piece appears? Answer like uv with a description.
14. LIATE letter for ln
The LIATE mnemonic orders u choices, and ln x heads the list. As one letter, which letter represents logarithms? Answer with an uppercase letter.