Wednesday, January 27, 2010

Q.8) Integration

Q : Evaluate ∫ (cos 7x - cos 8x) / (1 + 2 cos 5x) dx


SOLUTION :

Let I = ∫ (cos 7x - cos 8x) / (1 + 2 cos 5x) dx


Numerator of the integrand = (cos 7x - cos 8x) = 2 sin (15x/2) * sin (x/2)
Denominator of the integrand = (1 + 2 cos 5x) = cosec (5x/2) [ sin (5x/2) + 2 sin (5x/2) * cos (5x) ]
Therefore, I = ∫ (cos 7x - cos 8x) / (1 + 2 cos 5x) dx

=> I = ∫ (2 sin (15x/2) * sin (x/2) / cosec (5x/2) [ sin (5x/2) + 2 sin (5x/2) * cos (5x) ] dx
=> I = ∫ (2 sin (15x/2) * sin (5x/2) * sin (x/2) / [ sin (5x/2) + 2 sin (5x/2) * cos (5x) ] dx
=> I = ∫ (2 sin (15x/2) * sin (5x/2) * sin (x/2) / [ sin (5x/2) + sin (15x/2) - sin (5x/2) ] dx

=> I = ∫ (2 sin (15x/2) * sin (5x/2) * sin (x/2) / [sin (15x/2) ] dx
=> I = ∫ (2 sin (5x/2) * sin (x/2) dx
=> I = ∫ (cos 2x - cos 3x ) dx

=> (1/2) sin 2x - (1/3) sin 3x + c => (Ans)


 Link to Y/A

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