Calculus Volume 2 · 免费预览
部分分式
分母可分解的有理函数可以分解为若干简单分式之和，它们的积分是对数和反正切，从而把任何真有理函数的积分归结为已完成的积分表。
1. The decomposition theorem
The decomposition theorem
2. Long division first
Long division first
3. Irreducible quadratics and repeated factors
Irreducible quadratics and repeated factors
4. A full example
A full example
5. Proper fraction requirement
Before decomposing into partial fractions, a rational function must satisfy which condition?
6. Coefficient solving method
To find A and B in the decomposition of (3x + 5)/((x - 1)(x + 3)), which method is fastest?
7. Irreducible quadratic term
A partial fraction (Bx + C)/(x^2 + 4) integrates to which pair of terms?
8. Repeated linear factor
The denominator (x - 1)^2 contributes which partial fraction structure?
9. Heaviside cover-up
The shortcut of substituting each factor's root to isolate coefficients is known by which name?
10. Linear factor integral
The partial fraction A/(ax + b) integrates to what kind of function? Answer with one word.
11. Completing the square inside
A quadratic denominator x^2 + 2x + 5 is irreducible. Which completed form enables its partial fraction's integration?
12. Quadratic factor integral
The partial fraction over an irreducible quadratic integrates to a logarithm plus which other function? Answer with one word.
13. Division quotient
Dividing x^2 by x + 1 gives quotient x - 1 and remainder 1. How many terms does the decomposition of the proper remainder have over the single linear factor? Answer with a number.
14. Constant from substitution
Clearing denominators in 3x + 5 = A(x + 3) + B(x - 1) and substituting x = 1 gives 8 = 4A. What is A? Answer with a number.