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 x^{α - 1} e^{-β x^α}

Solution :

=  ∫[α β x^{α - 1} e^{-β x^α}] dx 

Let u = βx^α

du  =  αβ x^{(α - 1)} dx

∫[α β x^{α - 1} e^{-β x^α}] dx  =   ∫e^-u du

  =  - e^-u + c

  =  - e^(^-βx^α) + c

Question 2 :

Integrate the following with respect to x

tan x √sec x

Solution :

=  ∫[tan x √sec x] dx 

Let u = √sec x

u²  =  sec x

2u du  =  sec x tan x dx

2u du/sec x  =  tan x dx

∫[tan x √sec x] dx  =  ∫(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)¹⁷] dx 

Let u  =  (1 - x)

x  =  1 - u

dx  =  -du

∫[x (1 - x)¹⁷] dx  =  ∫(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 cos³x] dx 

=  ∫[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

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