INTEGRATION LECTURE NOTES

INTEGRATION LECTURE NOTES



INTEGRATION
Standard integrals
Examples:
d/dx ( -cos x)=sin x
d/dx (ex)=ex + c
power rule for integration
ʃ2x3 dx = x4/2 +c
ʃ x3/5 dx = x8/5/(8/5) + c
constant multiple rule and sum rule for any constant
ʃ c(f(x)) = c ʃ f(x)
The integral of a constant times a function is equal to the constant times
the integral of the function.
That is: ʃ [f(x) + g(x)] dx = ʃ f(x) dx + ʃ g(x) dx
The integral of a sum is the sum of the integrals
Example:
Find
ʃ (3ex) + 2/x – 0.5x2 +sin x dx
solution:
3ʃex dx + 2(1/x)dx + 0.5(x2)dx +sin x dx
=3ex + 2In x – 1/6 (x3)-cos x + c

, INTEGRATION LECTURE NOTES



INTEGRATION TABLE


F(X) ʃ F(X)dx
Xn Xn+1/n+1
1 X+c
a ax +c
Sin x -cos x +c
Cos x Sin x +c
Sec2x Tan x + c
ex ex + c
1/x In x + c

Methods of integration
 By substitution
 By parts
 Using partial fractions
 By inspection
 By use of table
 By reducing formula


By substitution
Suppose we use a variable u=u(x) which can be differentiated to give
du/dx
du=(du/dx) dx
by substituting:
u= ax +b
suppose we want to find ʃ(x + 4)5 dx