INTEGRATION IN THE FORM OF AX PLUS B PRACTICE WORKSHEET

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

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

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

Question 4 :

(i)  1/cos² (px + a)

(ii)  1/sin² (L - m x)

Question 5 :

(i)  ∫ e^3x dx

(ii)  ∫e^(x+3) dx

(iii)  ∫ e^(3x+2) dx

(iv)  ∫e^(5-4x) dxv

(v) ∫ e^-x dx

Question 6 :

(i)  ∫(e^x+1)/e^x dx

(ii) ∫e^{x log 2} e^x dx

(iii)  ∫e^{x loga} e^x dx

Question 7 :

∫ 5x⁴ e^x⁵ dx

Question 8 :

∫ e^x/(5+e^x) dx

Question 9 :

∫ e^{tan x}/cos²x dx

Question 10 :

∫ e^√x/√x  dx

Question 11 :

∫ e^ sin^-1 x/√(1-x²)  dx

(1)  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

(2)  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

(3)  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

(4)  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

(5)  Solution :

(i)  ∫ e^3x dx  =  (1/3) e^3x + C

(ii)  ∫e^(x+3) dx  =  (1/1) e^(x+3) + C

=  (1/1) e^(x+3) + C

(iii)  ∫ e^(3x+2) dx  =  (1/3) e^(3x+2) + C

=  e^(3x+2)/3 + C 

(iv)  ∫e^(5-4x) dx  =  (1/-4) e^(5-4x) + C

=  -e^(5-4x)/4 + C

(v) ∫ e^-x dx  =  e^-x/(-1) + C

= - e^-x + C

(vi)  ∫ 1/e^(p+qx) dx  =  ∫ e^-(p+qx) dx

=  e^-(p+qx)/(-q) + C

=  -1/q e^(p+qx) + C

(6)  Solution :

(i)  ∫(e^x+1)/e^x dx

=  ∫ [ (e^x/e^x) + 1/e^x ] dx

=  ∫dx + ∫e^-x dx

=  x - e^-x + C

(ii) ∫e^{x log 2} e^x dx

=  ∫e^log 2^x e^x dx

=  ∫2^x e^x dx  

=  ∫(2e)^x dx

=  (2e)^x/log ₂e + C

(iii)  ∫e^{x loga} e^x dx

=  ∫e^log a^x e^x dx

=  ∫a^x e^x dx  

=  ∫(ae)^x dx

=  (ae)^x/log _ae + C

(7)  Solution :

∫ 5x⁴ e^x⁵ dx

Let u  =  x⁵

du  =  5x⁴ dx

=  ∫ e^u du

=  e^u + C

=  e^x⁵ + C

(8)  Solution :

∫ e^x/(5+e^x) dx

Let u  =  5+e^x

du  =  e^x dx

∫ e^x/(5+e^x) dx  =  ∫ du/u

=  ln u + C

=  ln (5+e^x) + C

(9)  Solution :

∫ e^{tan x}/cos²x dx

1/cos²x  =  sec²x

∫ e^{tan x}/cos²x dx  =  ∫sec²x e^{tan x} dx

Let u  =  tanx

du  =  sec²x dx

=  ∫e^u dx

=  e^u + C

=  e^{tan x} + C

(10)  Solution :

∫ e^√x/√x  dx

Let √x  =  t

1/2√x dx  =  dt

1/√x dx  =  2 dt

=  ∫ e^t (2dt)

=  2∫ e^t dt

=  2e^t + C

=  2e^√x + C

(11)  Solution :

∫ e^ sin^-1 x/√(1-x²)  dx

Let u  =  sin^-1 x

du  =  1/√(1-x²) dx

∫ e^ sin^-1 x/√(1-x²)  dx  =  ∫ e^u du

=  e^u + C

=  e^sin^-1 x + C

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