Calculus Volume 1 · 免费预览
换元法
换元法是链式法则的逆过程，通过替换内层表达式及其微分，把复杂积分化为积分表中的基本条目，对不定积分和定积分都适用。
1. The method
The method
2. Definite integrals
Definite integrals
3. When algebra must prepare the way
When algebra must prepare the way
4. Connections and checks
Connections and checks
5. Substitution prerequisite
Which condition must hold for the substitution u = g(x) to convert an integral directly into a basic table entry?
6. Choice of u
For the integral of x^2 sqrt(x^3 + 1) dx, which substitution choice converts it into a pure power rule integral?
7. Definite limit transformation
In the definite integral from 0 to 1 of 2x (x^2 + 1)^3 dx with u = x^2 + 1, what are the new limits of integration?
8. Tangent integral
The integral of tan x dx is evaluated by rewriting tan x as sin x over cos x. Which substitution completes it?
9. Verification habit
After finding an indefinite integral by substitution, what is the most reliable check of the answer?
10. Which substitution fails
For which integrand would the substitution u = x^3 + 1 NOT directly apply?
11. Simple substitution value## practice-fill Simple substitution value
With u = x^2 + 1, the integral from 0 to 1 of 2x (x^2 + 1) dx becomes the integral of u du from 1 to 2. What is its value? Answer with a fraction like 3/2.
12. Constant balancing
The integral of x e^(x^2) dx uses u = x^2 with du = 2x dx. What constant factor must sit outside the transformed integral? Answer with a fraction like 1/2.
13. Logarithmic integral result
The integral of 1/(x ln x) dx with u = ln x becomes the integral of du/u. What is the result? Answer like ln|u| + C.
14. Tangent integral form
The integral of tan x dx equals the negative logarithm of which function, plus C? Answer with the function like cos x.