Integrate the following with respect to the respective variable : cos 3x cos 2x cos x - Mathematics and Statistics

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Sum

Integrate the following with respect to the respective variable : cos 3x cos 2x cos x

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Solution

Let I = `int cos 3x cos 2x cos x *dx`

Consider cos 3x cos 2x cos x = `(1)/(2) cos 3x [2 cos 2x cos x]`

= `(1)/(2)cos3x [cos(2x + x) + cos(2x - x)]`

= `(1)/(2)[cos^2 3x + cos3x cosx]`

= `(1)/(4)[2cos^2 3x + 2cos 3x cosx]`

= `(1)/(4)[1 + cos6x + cos(3x + x) + cos(3x - x)]`

= `(1)/(4)[1 + cos6x + cos4x + cos2x]`

∴ I = `(1)/(4) int[1 + cos6x + cos4x + cos2x]*dx`

= `(1)/(4) int 1*dx + 1/4 int cos6x*dx + 1/4 int cos4x*dx + 1/4 int cos2x*dx`

= `x/(4) + (1)/(4)((sin6x)/6) + 1/4((sin4x)/4) + 1/4((sin2x)/2) + c`

= `(1)/(48)[12x + 2sin 6x + 3sin 4x + 6sin2x] + c`.

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Chapter 3: Indefinite Integration - Miscellaneous Exercise 3 [Page 150]

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Balbharati Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board
Chapter 3 Indefinite Integration
Miscellaneous Exercise 3 | Q 2.7 | Page 150

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