INTEGRATION BY USING SUBSTITUTION METHOD

INTEGRATION BY USING SUBSTITUTION METHOD

 f [g(x)]g , (x)dx   f (u) du where u  g(x).

Steps of substitution method of integration:

Choose u  g(x) Substitute u  g(x) and Integrate Replace u
du  g '(x)dx to obtain with respect by g(x) in
and differentiate u to
get du  g '(x)dx. the integral  f (u)du. to u. the result.

 Example 1: Use suitable substitution to evaluate  2x 1 x2 5 dx.

Step 1: u 1 x2
du  2x  du  2xdx
Choose u, the differentiation of u dx
part of the integrand.

Step 2:

Substitute u  g(x) and   2x 1 x2 5 dx  u5du

du  g '(x)dx to obtain  f (u)du.

Step 3:  u6 c
Integrate with respect to u. 6
Step 4:
Replace u.   1 1 x2 6  c
6

Example 2: By using the substitution u  x  2 , find  3 dx
x2

Step 1: u x2

Choose u, the differentiation of u du  1  du  dx
part of the integrand. dx

Step 2:  3 2 dx   3 du
Substitute u  g(x) and x u

du  g '(x)dx to obtain  f (u)du. 1

  3u 2 du

Step 3: 1
Integrate with respect to u.
 6u 2  c

Step 4:  6x  1  c
Replace u.
22

Example 3: Use suitable substitution to evaluate the following 2xex23 dx

Step 1: u  x2  3

Chose correct u du  2x  du  2xdx
Differentiate u dx

Step 2:  2xe x23 dx  ex23 2x dx
Substitute u and du to obtain   eu du

 f (u)du.  eu  c

Step 3:  ex23  c
Integrate with respect to u.
Step 4:
Replace u.

Example 4: Use suitable substitution to evaluate the following  4x3 dx
x4  9

Step 1: u  x4  9

Chose correct u du  4x3  du  4x3dx
Differentiate u dx

Step 2:  4x3 dx  1 4 x 3 dx
Substitute u and du to obtain 
x4  9 x 4 9
 f (u)du.
  1 du
Step 3: u
Integrate with respect to u.
Step 4: Replace u.  ln u  c

 ln x4  9 c

Example 5: Use suitable substitution to evaluate the following  ln x 2 dx

x

Step 1: u  ln x

Chose correct u du  1  du  1 dx
Differentiate u dx x x

Step 2:  ln x 2 dx   u2 du
Substitute u and du to obtain
x
 f (u)du.
 u3 c
Step 3: 3
Integrate with respect to u.
Step 4:  ln x3  c
Replace u.
3

Example 6: Use suitable substitution to evaluate the following  x cos2x2 dx.

Step 1: u  2x2
Chose correct u du  4x  du  xdx
Differentiate u dx 4

Step 2:  x cos2x 2 dx   cos2x2 x dx
Substitute u and du to obtain
  cos u du
 f (u)du. 4

Step 3:  1  cos u du
Integrate with respect to u. 4
Step 4:
Replace u.   1 sin u  c
4

  1 sin 2x2  c
4