Page 82 - DMTH202_BASIC_MATHEMATICS_II

Unit 5: Definite Integrals by Substitution

 Notes
x
(d) 2 cos  cos sinx    dx
 
This problem not as awful as it appears. Here is the substitution and converted limits.
u  sin x du  cos x dx

 
x   u  sin  1 x    u  sin    0
2 2
The cosine in the very front of the integrand will get substituted away in the differential
and so this integrand in fact simplifies down considerably. Here is the integral.
 1
  2 cos  cos sinx   dx  0  cos u du
x
1
 sin   u
0

 sin   1  sin   0
 sin   1
Don’t get energized about these types of answers. On occasion we will finish up with trig
function assessments like this.
2
2 e w
(e) 1  dw
50 w 2
This is also a complicated substitution (at least until you see it). Here it is,
2 2 1 1
u  du   dw  dw   du
w w 2 w 2 2
1
w  2  u  1 w   u  100
50
Here is the integral.
2
2 e w 1 1 u
1  2 dw   100  e du
50 w 2
1
1
  e u
2 100
1
1
  e  e 100 
2
2 x dx
Task Evaluate  1 2 3 by using substitution method.
x   2
Self Assessment

Fill in the blanks:

6. The first method used to evaluate definite integral by substitution is to calculate the
indefinite integral, articulating an antiderivative in terms of the original variable, and
then assess the consequence at the ............................... limits.
7. The second method used to evaluate definite integral by substitution is to ...............................
the original limits to new limits in provisions of the new variable and do not translate the
antiderivative back to the original variable.
8. Sometimes a ............................... will stay similar after the substitution.

LOVELY PROFESSIONAL UNIVERSITY 77