INTEGRATION BY SUBSTITUTION  EXAMPLES WITH SOLUTIONS

About "Integration by Substitution Examples With Solutions"

Integration by Substitution Examples With Solutions :

Here we are going to see how we use substitution method in integration.

The method of substitution in integration is similar to finding the derivative of function of function in differentiation. By using a suitable substitution, the variable of integration is changed to new variable of integration which will be integrated in an easy manner.

Integration by Substitution Examples With Solutions - Practice Questions

Question 1 :

Integrate the following with respect to x

α β xα - 1 e-β x^α

Solution :

=  ∫[α β xα - 1 e-β x^αdx 

Let u = βxα

du  =  αβ x(α - 1) dx

∫[α β xα - 1 e-β x^αdx  =   ∫e-u du

  =  - e-u + c

  =  - e^(-βxα) + c

Question 2 :

Integrate the following with respect to x

tan x √sec x

Solution :

=  ∫[tan x √sec xdx 

Let u = sec x

u2  =  sec x

2u du  =  sec x tan x dx

2u du/sec x  =  tan x dx

∫[tan x √sec xdx  =  ∫(2u du/sec x) u

  =  ∫(2u/u2) u du

  =  2 ∫du

=  2 u + c

=  2  sec x + c

Question 3 :

Integrate the following with respect to x

x (1 - x)17

Solution :

=  ∫[x (1 - x)17dx 

Let u  =  (1 - x)

x  =  1 - u

dx  =  -du

∫[x (1 - x)17dx  =  ∫(1-u) u17 (-du) 

 =  - (u17 - u18) du

 =  - [u18/18 - u19/19]

 =  ((1 - x)19/19) - ((1 - x)18/18) + c

Question 4 :

Integrate the following with respect to x

sin5x cos3x

Solution :

=  ∫[sin5x cos3xdx 

=  ∫[sin5x cos2x cos x] dx 

=  ∫[sin5x (1-sin2x) cos x] dx 

Let u = sin x

du  =  cos x dx

∫[sin5x (1-sin2x) cos x] dx =  ∫u5 (1-u2) du 

  =  ∫(u5 - u7) du 

  =  u6/6 - u8/8 + c

  =  (sin6x/6) - (sin8x/8) + c

Question 5 :

Integrate the following with respect to x

cos x/cos (x-a)

Solution :

=  ∫[cos x/cos (x-a)dx 

Let u = x - a

du  =  dx 

x  =  u + a

cos x   =  cos (u + a)

cos x/cos (x-a)  =  cos (u + a) / cos u

  =  (cos u cos a + sin u sin a) / cos u

  =  cos a + tan u sin a

  =  ∫(cos a + tan u sin a) du

  =  cos a u + log sec u sin a  + c

  =  cos a (x - a) + log sec (x - a) sin a  + c

  =  (x - a) cos a + sin a log sec (x - a) + c

After having gone through the stuff given above, we hope that the students would have understood, "Integration by Substitution Examples With Solutions"

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