Indefinate Integration
Integration Formulae
Integration By Part
Algebric Integral
Solved Examples
Integration
Integration is the
reverse process of differentiation. The process of finding ƒ(x), when derivative
ƒ'(x) is given is known as integration .
Integral as antiderivative
Example
ƒ(x) = 4x
2 + 6
d/dx {ƒ(x)} = 8x
∫8xdx = 4x
2 +
c
where c is unknown constant.
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Differentiation Formulae
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If the integrand can be expressed as the product of two fuctions, then we use the following formula .
∫ƒ1(x)ƒ2(x)dx = ƒ1(x) ∫ƒ2(x)dx - ∫{(ƒ1'(x)
∫ƒ2(x)dx)}dx
where ƒ
1(x) and ƒ
2(x) are known as first and second function respectively .
In words: Integral of the product of two functions
Note : Order of ƒ
1(x) and ƒ
2(x) is normally decided by the rule
ILATE ,
where
I =
Inverse trigonometric
L =
Logarithmic
A =
Algebraic
T =
Trigonometric
E =
Exponential
Example : ∫1/(8x+2)dx .
Solution Putting 8x+2 = t, we get
8dx = dt
dx = (1/8)dt
Therefore, ∫1/(8x+2)dx = 1/8∫(1/t)dt
= 1/8 log t + C
= 1/8 log 8x+2 + C [Putting t = 8x+2]
Example :
Solution :
dx = xdt
Example : Integrate the function sinx.sin2x.sin3xdx
Solution : ∫sinx.sin2x.sin3xdx
∫(sinx.sin3x)sin2xdx
∫1/2[cos2x - cos4x]sin2xdx
∫1/2[cos2xsin2x - cos4xsin2x]dx
1/4 ∫[sin4x - (sin6x - sin2x)]dx
= 1/4 [∫sin4xdx - ∫sin6xdx + ∫sin2xdx]
Example : ∫xsinxdx
Solution : Let the first function x = ƒ
1(x) and
The second function = sinx = ƒ
2(x)
∫ƒ1(x)ƒ2(x)dx = ƒ1(x) ∫ƒ2(x)dx - ∫{(ƒ1'(x)
∫ƒ2(x)dx)}dx
∫xsinxdx = x ∫sinxdx - [∫(d/dx)x∫sinx dx]dx
= x(-cosx) - ∫[1.(-cosx)]dx
= -xcosx - ∫(-cosx)dx
= -xcosx + sinx + C
Example : Evaluate
Solution :
[Add and subtract 4 to convert x
2 + 4x into a perfect square]
Example : Integrate
Solution :
This is now easy to integrate:
