INTEGRATING TRIGONOMETRIC FUNCTIONS IN THE FORM OF AX PLUS B

Question 1 :

(i)  ∫cosec (2-x) cot (2-x) dx

(ii)  ∫cosec (4x+2) cot (4x+2) dx

(iii)  ∫cosec (3-2x) cot (3-2x) dx

(iv)  ∫cosec (Lx+m) cot (Lx+m) dx

Solution :

(i) 

∫cosec (2-x) cot (2-x) dx  =  -(1/-1) cosec (2-x) + C

=  cosec (2 - x) + C

(ii)

∫cosec (4x+2) cot (4x+2) dx  =  -(1/4) cosec (4x+2) + C

= -cosec (4x+2)/4 + C

(iii)

∫cosec (3-2x) cot (3-2x) dx  =  -(1/-2) cosec (3-2x) + C

=  cosec (3-2x)/2 + C

(iv)

∫cosec (Lx+m) cot (Lx+m) dx  = -(1/L) cosec (Lx+m) + C

= -cosec (Lx+m)/L + C

Question 2 :

(i)  ∫ sec (3+x) tan (3+x) dx

(ii)  ∫ sec (3x+4) tan (3x+4) dx

(iii)  ∫ sec (4-x) tan (4-x) dx

(iv)  ∫ sec (4-3x) tan (4-3x) dx

(v)  ∫sec (ax+b) tan (ax+b) dx

Solution :

(i)

∫ sec (3+x) tan (3+x) dx  =  (1/1) sec (3-x) + C

=  sec (3-x) + C

(ii)

∫ sec (3x+4) tan (3x+4) dx  =  (1/3) sec (3x+4) + C

=  sec (3x+4)/3 + C

(iii)

∫ sec (4-x) tan (4-x) dx  =  (1/-1) sec (4-x) + C

= - sec (4-x) + C

(iv) 

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

= - sec (4-3x)/3 + C

(v)  

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

Question 3 :

(i)  ∫cosec (2-x) cot (2-x) dx

(ii)  ∫cosec (4x+2) cot (4x+2) dx

(iii)  ∫cosec (3-2x) cot (3-2x) dx  

(iv)  ∫cosec (Lx+m) cot (Lx+m) dx

Solution :

(i)

∫cosec (2-x) cot (2-x) dx  = -(1/-1) cosec (2-x) + C

=  cosec (2-x) + C

(ii) 

∫cosec (4x+2) cot (4x+2) dx  =  -(1/4) cosec (4x+2) + C

=  -cosec (4x+2)/4 + C

(iii)

∫cosec (3-2x) cot (3-2x) dx  = -(1/-2) cosec (3-2x) + C

=  cosec (3-2x)/2 + C

(iv) 

∫cosec (Lx+m) cot (Lx+m) dx  =  -(1/L) cosec (Lx+m) + C

= - cosec (Lx+m)/L + C

Question 4 :

(i)  1/cos² (px + a)

(ii)  1/sin² (L - m x)

Solution :

(i)

∫ 1/cos² (px+a) dx  =  ∫sec²(px+a) dx

∫sec²(px+a) dx  =  (1/p) tan (px+a) + C

=  tan (px+a)/p + C

(ii)

∫ 1/sin² (L-mx) dx  =  ∫cosec² (L-mx) dx

∫cosec² (L-m x) dx  =  - (1/-m) cot (L-mx) + C

=  cot (L-mx)/m + C

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