INTEGRATING TRIGONOMETRIC FUNCTIONS USING SUBSTITUTION

Problem 1 :

Integrate cos¹⁴ x sin x

Solution :

t  =  cos x

Differentiate with respect to x

dt  =  -sin x dx

sin x dx  =  -dt

=  ∫cos¹⁴ x sin x  dx

=  ∫t¹⁴  (-dt)

=  t⁽¹⁴⁺¹⁾/(14+1) + C

=  t¹⁵/15 + C

=  (1/15) sin¹⁵ x + C

Problem 2 :

Integrate sin⁵ x

Solution :

=  ∫sin⁵ x  dx

=  ∫sin⁴ x sin x  dx

=  ∫(sin²x)² sin x  dx

=  ∫(1 - cos²x)² sin x  dx

t   =  cos x

dt  =  -sin x dx

sin x dx  =  - dt

=  ∫ (1-t²)² (-dt)

=  ∫ (1-t⁴-2t²) (-dt)

=  ∫(t⁴+2t²-1) dt

=  ∫t⁴ dt + ∫2t² dt - ∫1 dt

=  t⁵/5 + 2t³/3 - t + C

=  (1/5)cos⁵x + (2/3)cos³x - cos x + C

Problem 3 :

Integrate cos⁷x

Solution :

=  ∫ cos⁷x dx

=  ∫cos⁶x cos x   dx

=  ∫(cos² x)³ cos x  dx

=  ∫ (1-sin²x)³ cos x  dx

t  =  sin x

dt  =  cos x dx

=  ∫ (1-t²)³ dt

(a - b)³  =  a³-3a² b+3ab²-b³

=  ∫ [(1-3t²+3t⁴-t⁶] dt

=  t-3t³/3 + 3t⁵/5- t⁷/7 + C

=  t - t³+ 3t⁵/5- t⁷/7 + C

=  sin x - sin³x+ (3/5)sin⁵x - (1/7) sin⁷x + C

Problem 4 :

Integrate (1 + tan x)/(x + log sec x)

Solution :

t  =  (x+log sec x)

dt  =  1 + (1/sec x) sec x tan x

dt  =  1+tan x

=  ∫dt/t

=  log t + C

=  log (x + log sec x) + C

Problem 5 :

Integrate e^(m tan^-1x)/(1+x²)

Solution :

t  =  tan^-1x

dt  =  1/(1+x²) dx

=  ∫e^(m tan^-1x)/(1+x²) dx

=  ∫e^mt dt

=  e^mt/m + C

=  e^(m tan^-1x)/m + C

Problem 6 :

Integrate x sin^-1 (x²)/√(1-x⁴)

Solution :

t  =  sin^-1 (x²)

dt  =  [1/√(1 - (x²)²] (2 x) dx

dt/2  =  [1/√(1 - x⁴] x dx

=  ∫x sin^-1 (x²)/√(1 - x⁴) dx

=  ∫ t (dt/2)

=  (1/2) ∫t dt

=  (1/2) t²/2 + C

=  (1/4) [sin^-1 (x²)]² + C

Problem 7 :

Integrate 5 (x + 1) (x + log x)⁴/x

Solution :

t  =  (x+log x)

differentiating with respect to x

dt  =  [1+(1/x)] dx

dt  =  [(x+1)/x)] dx

=  ∫5 (x+1) (x+log x)⁴/x dx

=  ∫5 t⁴ dt

= 5 t⁵/5 + C

=  t⁵ + C

=  (x + log x)⁵ + C

Problem 8 :

Integrate sin (log x)/x

Solution :

t  =  log x

differentiating with respect to "x"

dt  =  (1/x) dx

=  ∫sin (log x)/x dx

=  ∫sin t dt

=  cos t + C

=  cos (log x) + C 

Problem 9 :

Integrate cot x/log sin x

Solution :

t  =  log sin x

dt  =  (1/sin x)cos x dx

dt  =  cot x

=  ∫cot x/log sin x dx

=  ∫(1/t) dx

=  log t + C

=  log (log sin x) + C

Problem 10 :

Integrate sec⁴x tan x

Solution :

=  ∫ sec⁴xtan x dx

=  ∫sec³ x (sec x tan x) dx

t  =  sec x

differentiating with respect to x

dt  =  sec x tan x dx

=  ∫t³ dt

=  (t⁴/4) + C

=  (sec ⁴ x/4) + C

Problem 11 :

Integrate tan³x sec x

Solution :

=  ∫tan³x sec x dx

t  =  sec x

differentiating with respect to x

dt  =  sec x tan x dx

=  ∫tan ²x tan x sec x dx

= ∫(sec²x - 1) tan x sec x dx

=  ∫(sec³ x tan x - tan x sec x) dx

= ∫ sec² x sec x tan x dx - ∫ tan x sec x dx

= ∫ t² dt - ∫ dt

=  t³/3 dt - t + C

=  (sec³x/3) - sec x + C

Problem 12 :

Integrate sin x/sin (x + a)

Solution :

=  ∫sin x/sin (x+a) dx

=  ∫ sin (x+a-a)/sin (x + a) dx

sin (A- B) = sin A cos B - cos A sin B

=  ∫[sin (x+a) cos a - cos (x+a) sin a]/sin (x+a) dx

=  ∫[sin (x+a) cos a]/sin (x+a) dx - ∫[cos (x+a) sin a]/sin (x+a) dx

=  ∫cos a dx-∫cot (x+a) sin a dx

=  cos ax - sin a log sin (x+a) + C

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