Important Definite Integrals Formulas for JEE and NEET

 Page 1


Page # 17
DEFINITE INTEGRA TION
Properties of definite integral
1.  
?
b
a
) x ( f dx = 
?
b
a
) t ( f dt 2. 
?
b
a
) x ( f
 dx = – 
?
a
b
) x ( f
dx
3. 
?
b
a
) x ( f
 dx = 
?
c
a
) x ( f dx + 
?
b
c
) x ( f dx
4. 
?
?
a
a
) x ( f dx  = 
?
? ?
a
0
)) x ( f ) x ( f (
 dx  =  
?
?
?
?
?
?
?
?
)x ( f – )x (– f , 0
)x ( f ) x (– f , dx )x ( f 2
a
0
5.  
?
b
a
) x ( f dx  = 
?
? ?
b
a
) x b a ( f dx
Page 2


Page # 17
DEFINITE INTEGRA TION
Properties of definite integral
1.  
?
b
a
) x ( f dx = 
?
b
a
) t ( f dt 2. 
?
b
a
) x ( f
 dx = – 
?
a
b
) x ( f
dx
3. 
?
b
a
) x ( f
 dx = 
?
c
a
) x ( f dx + 
?
b
c
) x ( f dx
4. 
?
?
a
a
) x ( f dx  = 
?
? ?
a
0
)) x ( f ) x ( f (
 dx  =  
?
?
?
?
?
?
?
?
)x ( f – )x (– f , 0
)x ( f ) x (– f , dx )x ( f 2
a
0
5.  
?
b
a
) x ( f dx  = 
?
? ?
b
a
) x b a ( f dx
Page # 18
6.  
?
a
0
) x ( f dx = 
?
?
a
0
) x a ( f dx
7.  
?
a 2
0
) x ( f
 dx  = 
?
? ?
a
0
)) x a 2 (f ) x (f ( dx  = 
?
?
?
?
?
?
?
?
) x (f – ) x – a 2 (f , 0
)x ( f )x – a 2 ( f , dx )x ( f 2
a
0
8.  If f(x) is a periodic function with period T, then
     
?
nT
0
) x ( f dx  = n 
?
T
0
) x ( f dx, n ? z,   
?
?nT a
a
) x ( f dx  = n 
?
T
0
) x ( f dx, n ? z, a ? R
 
?
nT
mT
) x ( f dx  = (n – m) 
?
T
0
) x ( f dx, m, n ? z,   
?
?nT a
nT
) x ( f dx  = 
?
a
0
) x ( f dx, n ? z, a ? R
?
?
?
nT b
nT a
) x ( f dx  = 
?
a
a
) x ( f dx, n ? z, a, b ? R
9.     If ?(x) ? f(x) ? ? (x) for a ? x ? b,
         then   
?
?
b
a
) x (
 dx  ?  
?
b
a
) x ( f
 dx ?  
?
?
b
a
) x ( dx
10.  If m ? f(x) ? M for a ? x ? b, then m (b – a) ? 
?
b
a
) x ( f dx ? M (b – a)
11.  If f(x) ? 0 on [a, b] then 
?
b
a
dx ) x ( f ? 0
Leibnitz Theorem :   If F(x) = 
?
) x ( h
) x ( g
dt )t (f , then
dx
) x ( dF
 = h ?(x) f(h(x)) – g ?(x) f(g(x))