Integration
Integration (Anti-Differentiation)
The process of finding the function of a given derivative.
Integral
The integral gives the area under the curve of a function.
Indefinite Integral
If φ´(x) = f(x), then
φ(x) + c is an indefinite integral of f(x) and is denoted as ∫ f(x) dx = φ(x) + c.
Integrals of Standard Functions
∫ dx=   x + c
∫ k dx=   kx + c ,   where k is constant
∫ k f(x) dx=   k ∫ f(x) dx ,   where k is constant
∫ x dx=   x2 2 + c
∫ xn dx=   xn + 1 n + 1 + c ,   where n ≠ -1
∫ (ax + b) dx=  
1 a
(ax + b) + c
∫ (ax + b)n dx=  
(ax + b)n + 1 a (n + 1)
+ c ,   where n ≠ -1
∫
1 x
dx		=   ln x + c
∫
1 ax + b
dx		=  
1 a
ln ax + b + c
∫ ex dx=   ex + c
∫ ax dx=  
ax ln(a)
+ c
=   ax logae + c
∫ ln(x) dx=   x ln(x) − x + c
∫ x dx=  
x3/2 3/2
∫
1 a2 − x2
dx		=   sin-1
x a
+ c
=   − cos-1
x a
+ c
∫
1 x2 + a2
dx		=   ln x + x2 + a2 + c
∫
1 x2 − a2
dx		=   ln x + x2 − a2 + c
∫
1 a2 + x2
dx		=  
1 a
tan-1
x a
+ c
=   −
1 a
cot-1
x a
+ c
∫
1 a2 − x2
dx		=  
1 2a
log
a + x a − x
+ c
∫
1 x2 − a2
dx		=  
1 2a
log
x − a x + a
+ c
Integrals of Trigonometric Functions
∫ sin(x) dx=   − cos(x) + c
∫ cos(x) dx=   sin(x) + c
∫ tan(x) dx=   − ln cos(x) + c
∫ cot(x) dx=   ln sin(x) + c
∫ sec(x) dx=   ln sec(x) + tan(x) + c
∫ cosec(x) dx=   − ln cosec(x) + cot(x) + c
∫ sec2(x) dx=   tan(x) + c
Integrals of Inverse Trigonometric Functions
∫ sin-1(x) dx=   x sin-1(x) + 1 - x2 + c
∫ cos-1(x) dx=   x cos-1(x) + 1 - x2 + c
∫ tan-1(x) dx=   x tan-1(x) −
ln(1 + x2) 2
+ c
∫ cot-1(x) dx=   x cot-1(x) +
ln(1 + x2) 2
+ c
∫ sec-1(x) dx=   x sec-1(x) − ln x + x x2 − 1 x2 + c
∫ cosec-1(x) dx=   x cosec-1(x) + ln x + x x2 − 1 x2 + c
Integrals of Hyperbolic Functions
∫ sinh(x)=   cosh(x) + c
∫ cosh(x)=   sinh(x) + c
∫ tanh(x)=   ln cosh(x) + c
∫ coth(x)=   ln sinh(x) + c
∫ sech(x)=   tan-1sinh(x) + c
∫ cosech(x)=   ln tanh(x/2) + c
Rules of Integration
Sum Rule
∫(u + v)dx   =   ∫ u dx + ∫ v dx
Difference Rule
∫(u − v)dx   =   ∫ u dx − ∫ v dx
Product Rule
∫ku dx  =   k ∫ u dx ,   where k is a constant
Integration by Parts
∫(uv) dx  =   u ∫ v dx − ∫ [
du dx
∫ v dx ] dx
Integration by Substitution
1.
If x = φ(t) is a differential function of t, then:
∫ f(x) dx   =   ∫ f[φ(t)] φ´(t) dt
2.
If ∫ f(x) dx = φ(x), then:
∫ f(ax + b) dx   =  
φ(ax + b) a
+ c
3.
∫ [f(x)]n f´(x) dx   =  
[f(x)]n + 1 n + 1
+ c
4.
∫
f´(x) f(x)
dx   =   logf(x) + c
5.
∫
f´(x) n f(x)
dx   =  
[f(x)]1 - (1/n) 1 − (1/n)
+ c