COMPUTING INDEFINITE INTEGRALS PRACTICE PROBLEMS

Problem 1 :

∫x⁶ dx

Solution :

∫ xⁿ dx = x⁽ⁿ⁺¹⁾/(n + 1) + c

∫x⁶ dx  =  x⁽⁶⁺¹⁾/(6+1) + C

Problem 2 :

∫x^-2 dx

Solution :

∫x^-2 dx  =  x^(-2+1)/(-2+1) + C

=  x^-1/(-1) + C

=  1/x + C

Problem 3 :

∫ √x⁵ dx

Solution :

∫ √x⁵ dx  =  (x⁵)^1/2

  =  ∫x^5/2 dx

=  x^(5/2+1)/(5/2+1) + C

=  x^7/2/(7/2) + C

=  (2/7)x^7/2 + C

Problem 4 :

∫ sin x/cos²x dx

Solution :

∫sin x/cos ²x dx  =  ∫(sin x/cos x) (1/cos x) dx

=  ∫tan x sec x dx

=  sec x + c

Problem 5 :

∫ (cot x/sin x) dx

Solution :

∫ (cot x/sin x) dx = ∫(cot x) ⋅ (1/ sin x) dx

=  ∫cot x cosec x dx

=  - cosec x + c

Problem 6 :

∫ (1/sin² x) dx

Solution :

∫(1/sin²x) dx   = ∫ cosec² x dx

=  - cot x + c

Problem 7 :

∫ (3-4x)⁶ dx

Solution :

∫(3-4x)⁶ dx   =  (-1/4)(3-4x)⁽⁶⁺¹⁾/(6+1) + C

=  (-1/28) (3-4x)⁷ + C

Problem 8 :

∫ 1/(3+5x) dx

Solution :

∫ 1/(3+5x) dx   =  log (3+5x)/5 + C

=  (1/5) log(3+5x) + C

Problem 9 :

∫cosec (4x+3) cot (4x+3) dx

Solution :

∫cosec (4x+3) cot (4x+3) dx  =  -cosec (4x+3) (1/4) + C

=  - (1/4) cosec (4x+3) + C 

Problem 10 :

∫sec (ax+b) tan (ax+b) dx

Solution :

∫ ∫sec (ax+b) tan (ax+b) dx  =  sec (ax+b) (1/a) + C

=  (1/a) sec (ax+b) + C

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