Begin2.DVI

Appendix C

Table of Integrals

Indefinite Integrals

General Integration Properties

1. IfdFdx(x)=f(x), then

∫

f(x)dx=F(x) +C

2. If

∫

f(x)dx=F(x) +C, then the substitutionx=g(u)gives

∫

f(g(u))g′(u)du=F(g(u)) +C

For example, if

∫ dx

x^2 +β^2 =

1

βtan

− 1 x

β+C, then

∫ du

(u+α)^2 +β^2 =

1

βtan

− 1 u+α

β +C

3. Integration by parts. Ifv 1 (x) =

∫

v(x)dx, then

∫

u(x)v(x)dx=u(x)v 1 (x)−

∫

u′(x)v 1 (x)dx

4. Repeated integration by parts or generalized integration by parts.
   Ifv 1 (x) =

∫

v(x)dx, v 2 (x) =

∫

v 1 (x)dx,... , vn(x) =

∫

vn− 1 (x)dx, then

∫

u(x)v(x)dx=uv 1 −u′v 2 +u′′v 3 −u′′′v 4 +···+ (−1)n−^1 un−^1 vn+ (−1)n

∫

u(n)(x)vn(x)dx

5. Iff−^1 (x)is the inverse function off(x)and if

∫

f(x)dxis known, then

∫

f−^1 (x)dx=zf(z)−

∫

f(z)dz, where z=f−^1 (x)

6. Fundamental theorem of calculus.
   ∫If the indefinite integral off(x)is known, say
   f(x)dx=F(x) +C, then the definite integral

∫b

a

dA=

∫b

a

f(x)dx=F(x)]ba=F(b)−F(a)

represents the area bounded by the x-axis, the curve

y = f(x)and the linesx=aandx=b.

7. Inequalities.
   (i) Iff(x)≤g(x)for allx∈(a, b), then

∫b

a

f(x)dx≤

∫b

a

g(x)dx

(ii) If|f(x)≤M|for allx∈(a, b)and

∫b

af(x)dxexists, then

∣∣

∣∣

∣

∫b

a

f(x)dx

∣∣

∣∣

∣≤

∫b

a

f(x)dx≤M(b−a)

Appendix C