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672
Mathematical Techniques, Identities, and Tables
Appendix A
Integration Techniques
1. Change in variable. Let v = u(x):
b
u(b)
f(x) dx = a f(x)
La
L u(a) dv>dx ` b dv
x = u -1 (v)
3. Integral tables.
4. Complex variable techniques.
5. Numerical methods.
2. Integration by parts
L u dv = uv - L v du
A–5 INTEGRAL TABLES
Indefinite Integrals
L (a + (a + bx)n+1
bx)n dx = , 0 6 n
b(n + 1)
dx
La + bx = 1
ln ƒ a + bxƒ
b
dx
L(a + bx) n = -1
(n - 1)b(a + bx) n-1 , 1 6 n
2
24ac - b 2 tan-1 ¢
2ax + b
24ac - b ≤, 2
dx
L c + bx + ax 2 = 1 2ax + b - 2b
g
2 - 4ac
In `
2b 2 - 4ac 2ax + b + 2b 2 - 4ac ` ,
-2
22ax + b ,
x dx
L c + bx + ax 2 = 1
2a ln ƒ ax2 + bx + c ƒ -
dx
L a 2 + b 2 x 2 = 1
ab tan-1 a bx a b
x dx
L a 2 + x 2 = 1 2 ln (a2 + x 2 )
b dx
2a L c + bx + ax 2
b2 6 4ac
b2 7 4ac
b2 = 4ac
cos x dx = sin x
L
L
x cos x dx = cos x + x sin x
L
sin x dx = -cos x
x sin x dx = sin x - x cos x
L