INTEGRATION PRACTICE WORKSHEET

Question 1 :

Integrate the following with respect to x

∫ [(x + 4)⁵ + 5/(2 - 5x)⁴ - cosec²(3x - 1)] dx

Solution :

∫ [(x + 4)⁵ + 5/(2 - 5x)⁴ - cosec²(3x - 1)] dx

=  ∫ (x + 4)⁵ dx +  ∫(5/(2 - 5x)⁴) dx -  ∫cosec²(3x - 1) dx

=  (x + 4)⁶/6 +  5∫(2 - 5x)^-4 dx -  ∫cosec²(3x - 1) dx

=  (x + 4)⁶/6 +  5(2-5x)^-3/(-3)(-5) dx -  cot(3x-1)/(-3) + c

=  (1/6)(x + 4)⁶ - (1/3)(1/(2-5x)³) + (1/3) cot(3x-1) + c

Question 2 :

Integrate the following with respect to x

∫ [4 cos (5-2x) + 9e^3x-6 + 24/(6-4x)] dx

Solution :

∫ [4 cos (5-2x) + 9e^3x-6 + 24/(6-4x)] dx

=  ∫ 4 cos (5-2x) dx + 9 ∫ e^3x-6dx + ∫24/(6-4x) dx

=  4 sin (5 - 2x)/(-2) + 9 e^3x-6/3 + 24 log (6 - 4x)(-1/4) + c

=  -2 sin (5 - 2x) + 3 e^3x-6 - 6 log (6 - 4x) + c

Question 3 :

Integrate the following with respect to x

∫ [sec²(x/5) + 18 cos 2x  + 10 sec (5x + 3) tan (5x + 3)] dx

Solution :

∫ [sec²(x/5) + 18 cos 2x  + 10 sec (5x + 3) tan (5x + 3)] dx

= ∫sec²(x/5)dx+18∫cos 2xdx+10∫sec(5x+3)tan(5x+3)] dx

=  tan (x/5)/(1/5)+(18/2)sin 2x+(10/5) sec (5x+3) + c

=  5 tan (x/5) + 9 sin 2x + 2 sec (5x+3) + c

Question 4 :

Integrate the following with respect to x

∫ [8/√(1 - (4x)²) + 27/√(1 - 9x²) - 15/(1+25x²)] dx

Solution :

  =  ∫8/√(1-(4x)²) dx+∫27/√(1-9x²) dx-∫15/(1+25x²) dx

  =  8∫1/√(1-(4x)²)dx+27∫1/√(1-(3x)²) dx-15∫1/(1+(5x)²) dx

=  (8/4)sin^-1(4x) + (27/3) sin^-1(3x) - (15/5) tan^-1(5x) + c 

=  2 sin^-1(4x) + 9 sin^-1(3x) - 3 tan^-1(5x) + c 

Question 5 :

Integrate the following with respect to x

∫ [6/(1 + (3x + 2)²)] - [12/√1 - (3-4x)²] dx

Solution :

=  ∫ [6/(1 + (3x + 2)²)] - [12/√1 - (3-4x)²] dx

  =  6∫[1/(1 + (3x + 2)²)] - 12 ∫[1 √1 - (3-4x)²] dx

  =  (6/3) tan^-1(3x+2) - (12/(-4)) sin^-1(3-4x) + c

  =  2 tan^-1(3x+2) + 3 sin^-1(3-4x) + c

Question 6 :

Integrate the following with respect to x

∫ (1/3) cos ((x/3) - 4)) + 7/(7x+9) + e^{(x/5) + 3} dx

Solution :

=  ∫ (1/3) cos ((x/3) - 4)) dx + ∫7/(7x+9) dx + ∫e^(x/5)+3 dx

= (1/3)/(1/3)sin((x/3)-4))+(7/7)log(7x+9) + (1/5)e^{(x/5) + 3}+c

= (1/3)/(1/3)sin((x/3)-4))+log(7x+9) + 5e^{(x/5) + 3}+c 

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