Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/20

1.  

   CHAPTER XXV
   INTEGRATION OF RATIONAL FRACTIONS

2. SECTION

   PAGE

3. 184.

   Introduction ................................................................................................................................................................................................................................................................................................................................................................................................

   325

4. 185.

   Case I ................................................................................................................................................................................................................................................................................................................................................................................................

   325

5. 186.

   Case II ................................................................................................................................................................................................................................................................................................................................................................................................

   327

6. 187.

   Case III ................................................................................................................................................................................................................................................................................................................................................................................................

   329

7. 188.

   Case IV ................................................................................................................................................................................................................................................................................................................................................................................................

   331

8.  

   CHAPTER XXVI
   INTEGRATION BY SUBSTITUTION OF A NEW VARIABLE. RATIONALIZATION

9. 189.

   Introduction ................................................................................................................................................................................................................................................................................................................................................................................................

   335

10. 190.

    Differentials containing fractional powers of $\scriptstyle {x}$ only ................................................................................................................................................................................................................................................................................................................................................................................................

    335

11. 191.

    Differentials containing fractional powers of $\scriptstyle {a+bx}$ only ................................................................................................................................................................................................................................................................................................................................................................................................

    336

12. 192.

    Change in limits corresponding to change in variable ................................................................................................................................................................................................................................................................................................................................................................................................

    336

13. 193.

    Differentials containing no radical except $\scriptstyle {\sqrt {a+bx+x^{2}}}$ ................................................................................................................................................................................................................................................................................................................................................................................................

    338

14. 194.

    Differentials containing no radical except $\scriptstyle {\sqrt {a+bx-x^{2}}}$ ................................................................................................................................................................................................................................................................................................................................................................................................

    338

15. 195.

    Binomial differentials ................................................................................................................................................................................................................................................................................................................................................................................................

    340

16. 196.

    Conditions of integrability of binomial differentials ................................................................................................................................................................................................................................................................................................................................................................................................

    341

17. 197.

    Transformation of trigonometric differentials ................................................................................................................................................................................................................................................................................................................................................................................................

    343

18. 198.

    Miscellaneous substitutions ................................................................................................................................................................................................................................................................................................................................................................................................

    345

19.  

    CHAPTER XXVII
    INTEGRATION BY PARTS. REDUCTION FORMULAS

20. 199.

    Formula for integration by parts ................................................................................................................................................................................................................................................................................................................................................................................................

    347

21. 200.

    Reduction formulas for binomial differentials ................................................................................................................................................................................................................................................................................................................................................................................................

    350

22. 201.

    Reduction formulas for trigonometric differentials ................................................................................................................................................................................................................................................................................................................................................................................................

    356

23. 202.

    To find $\scriptstyle {\int e^{ax}\sin {nx}dx}$ and $\scriptstyle {\int e^{ax}\cos {nx}dx}$ ................................................................................................................................................................................................................................................................................................................................................................................................

    359

24.  

    CHAPTER XXVIII
    INTEGRATION A PROCESS OF SUMMATION

25. 203.

    Introduction ................................................................................................................................................................................................................................................................................................................................................................................................

    361

26. 204.

    The fundamental theorem of Integral Calculus ................................................................................................................................................................................................................................................................................................................................................................................................

    361

27. 205.

    Analytical proof of the Fundamental Theorem ................................................................................................................................................................................................................................................................................................................................................................................................

    364

28. 206.

    Areas of plane curves. Rectangular coördinates ................................................................................................................................................................................................................................................................................................................................................................................................

    365

29. 207.

    Area when curve is given in parametric form ................................................................................................................................................................................................................................................................................................................................................................................................

    368

30. 208.

    Areas of plane curves. Polar coördinates ................................................................................................................................................................................................................................................................................................................................................................................................

    370

31. 209.

    Length of a curve ................................................................................................................................................................................................................................................................................................................................................................................................

    372

32. 210.

    Lengths of plane curves. Rectangular coördinates ................................................................................................................................................................................................................................................................................................................................................................................................

    373

33. 211.

    Lengths of plane curves. Polar coördinates ................................................................................................................................................................................................................................................................................................................................................................................................

    375