∫sin(ax+b) dx = -(1/a) cos (ax+b) + C
∫cos(ax+b) dx = (1/a) sin (ax+b) + C
∫sec2 (ax+b) dx = (1/a) tan (ax+b) + C
∫sec (ax+b) tan (ax+b) dx = (1/a) sec (ax+b) + C
∫cosec2 (ax+b) dx = -(1/a) cot (ax+b) + c
∫cosec (ax+b) cot (ax+b) dx = -(1/a) cosec (ax+b) + c
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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