Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

114_Indefinite Integrals_[Ch. 4
If [g(u)du is known, that is,
then
Actually, we have already made use oi this method in Sec. 1,3.
Examples 2, 3, 4 (Sec. 1) may be solved as follows:
Example 2. u = 5# 2; cfw = 5c(x; dx -du.
1 d 1
, -~ _ + c
Example 3. u = x*; du 2xdx; xdx = - .
Example 4. w = jc s ; du = 3x 2 dx\ x 2 dx = .
'
_ 2
-
2. Trigonometric substitutions. _
1) If an integral contains the radical ]fa z
x 2
, the usual thin^ is to ~a sin /; whence
put
whence
2) If an integral contains the radical V* 2
/^x 2
a 2 = a tan ^.
3) If an integral contains the radical V'V + a 2
,
a 2
, we put xawct,
we put* = atan/; whence
It should be noted that trigonometric substitutions do not always turn
out to be advantageous.
It is sometimes more convenient to make use of hyperbolic substitutions,
which are similar to trigonometric substitutions (see Example 1209).
For more details about trigonometric and hyperbolic substitutions, see
Sec. 9.
Example
5. Find
i^dx.