Method 7The integration of any hyperbolic function.
(85)
(86)
(87)
(88)
(89)
(90)
Then (91)
(92)
Example(93)
(94)
Method 8Trigonometrical substitution Integration of irrational equations containing (95)
Example 1(96)
(97)
(98)
(99)
(100)
Example 2Find the integral of (101)
Let
(102)
(103)
(104)
(105)
(106)
(107)
(108)
Method 9Integration by parts (109)
(110)
(111)
this can also be written as:- (112)
Example 1(113)
(114)
(115)
Example 2(116)
(117)
(118)
Integration by parts twice to regain the original integration. (119)
(120)
(121)
(122)
(123)
Method 10A large number of expressions can only be integrated by the method of successive reductions. This consists of making the integral dependent on a simpler integral, then again reducing this to one simpler still until a known form is found.
Example(124)
(125)
(126)
(127)
(128)
(129)