1. 求下列有理函数的不定积分:
   1. `int (2x-1)/(x^2+3x+2) dx`;
   2. `int (x dx)/(x^3-3x+2)`;
   3. `int (x^3+1)/(x^3-5x^2+6x) dx`;
   4. `int dx/(x^3+x^2+x+1)`;
   5. `int dx/(x^4+2x^3+2x^2+x)`;
   6. `int dx/(x^3+1)`;
   7. `int ((2x-1)dx)/(x^4+x)`;
   8. `int (x^2+1)/(x^4+x^2+1) dx`;
   9. `int dx/(x^6+1)`;
   10. `int dx/(x^6-1)`;
   11. `int (x dx)/((x-1)^2 (x+1)^3)`;
   12. `int dx/(x^2+1)^3`.
2. 用尽可能简单的方法求下列有理函数的不定积分:
   1. `int dx/(x(x^10+2))`;
   2. `int dx/(x(x^10+1)^2)`;
   3. `int ((1-x^7) dx)/(x(1+x^7))`;
   4. `int (x^4+1)/(x(x^4+5)(x^5+5x+1)^2) dx`;
   5. `int ((1-x^7) dx)/(x(1+x^7))`;
   6. `int (x^4+1)/(x(x^4+5)(x^5+5x+1)^2) dx`;
   7. `int (x^5+x)/(x^8+1) dx`;
   8. `int (x^5-x)/(x^8+1) dx`;
   9. `int (x^4+1)/(x^6+1) dx`;
   10. `int (x^2 dx)/(x^3+x^2+x+1)`.
3. 设 `P_n(x)` 是 `n` 次多项式. 求积分 `int (P_n(x))/(x-a)^n dx`.
4. 设 `m, n` 为自然数. 应用变量代换 `u = (x-a)/(x-b)` 求积分 `int dx/((x-a)^m (x-b)^n)`.
5. 求计算积分 `I_n = int dx/(ax^2+bx+c)^n` (`a != 0`, `n = 1, 2, cdots`) 的递推公式.
6. 求下列三角函数有理式的不定积分:
   1. `int cos^5 x dx`;
   2. `int sin^6 x dx`;
   3. `int sin^4 x cos^5 x dx`;
   4. `int sin^5 x cos^5 x dx`;
   5. `int sin^2 x cos^4 x dx`;
   6. `int sin^3 2x cos^4 x dx`;
   7. `int sin^2 ax cos^3 bx dx`;
   8. `int sin ax cos bx cos cx dx`;
   9. `int (sin^3 x)/(cos^4 x) dx`;
   10. `int (sin^3 x)/(cos^5 x) dx`;
   11. `int (sin^4 x)/(cos^3 x) dx`;
   12. `int (sin^5 x)/(cos^3 x) dx`;
   13. `int dx/(cos^3 x)`;
   14. `int dx/(sin x cos^4 x)`;
   15. `int dx/(sin^4 x)`;
   16. `int dx/(sin^4 x cos^4 x)`;
   17. `int tan^4 x dx`;
   18. `int "cot"^5 x dx`.
7. 设 `n` 为自然数. 求计算积分 `I_n = int sin^n x dx` 和 `J_n = int cos^n x dx` 的递推公式.
8. 设 `n` 为自然数. 求计算积分 `I_n = int dx/(sin^n x)` 和 `J_n = int dx/(cos^n x)` 的递推公式.
9. 求下列无理函数的不定积分:
   1. `int dx/(1+root 3 x)`;
   2. `int (1-sqrt(x+1))/(1+root 3(x+1))dx`;
   3. `int (sqrt(x+1) - sqrt(x-1))/(sqrt(x+1) + sqrt(x-1)) dx`;
   4. `int (x^2 dx)/sqrt(x^2 + x + 1)`;
   5. `int (x^3 dx)/sqrt(1+2x-x^2)`;
   6. `int dx/(x^3 sqrt(x^2+1))`;
   7. `int (sqrt x dx)/(1 + root 3 x)^2`;
   8. `int dx/root 3 (1+x^3)`.