for n substitute n-2, and divide by (n−1)a2;

dx

dx

dx

n√ =

x" (a2 — x2)}

;

Finally, when n is even, the formula is applicable, and we have

;

Which formula is applicable for all values of n; and at last, when

= 2 x"− 1 ( a + bx + cx2) 1 — 2 (n − 1) f x"−2 (a + bx + cx2)3 dx

Substituting which in (99), and adding and reducing,

x dx (a + bx + cx2) 1

=

=

x+3

(2−2x+x2) 1 +

= √_d(x−1)

{(x-1)2+1}+

3

x dx

+

2 √ (2−2 x + x2) 12

dx

;

(2−2x+x2)1

= log {x−1+(2−2x+x2)1}.

x + 3 (2 − 2 x + x2) 3 + {} log {x−1+(2−2x+x2)1 }.

60.] Integration of a" ex dx, where n is a positive integer.

(1)

(2)

(3)

(4)

which does not admit of integration in finite terms, but may be

62.] The preceding are the integrals of the simpler exponential functions; other combinations however often admit of reduction to algebraic forms by means of substitution, and thereby of integration by the methods of the last Chapter; of these some examples are subjoined.