# Study the phenomena of resonance for an oscilating system where f(x)=sinx*cos2x

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We have to study resonance for f(x) = sin x * cos 2x

Now we need to find the integral of f(x).

We know that sin x * cos y = [sin (x-y) + sin (x +y) ]/2

Therefore f(x) = sin x * cos 2x

=> f(x) = (1/2) [ sin -x + sin 3x ]

For the integral

Int [ sin x * cos 2x ]

=> Int [ (sin (x- 2x) + sin (x+2x)) /2 dx]

=> (1/2) Int [sin (-x) + sin (3x) dx ]

=>(1/2) Int [ sin (-x)] + (1/2) Int [ sin 3x]

=> (1/2) [ cos x] - (1/2) [cos 3x /3] + C

=> (cos x) / 2 - (cos 3x) / 6 + C

To study the phenomena of resonance, we'll just have to calculate the indefinite integral of the given function:f(x)=sinx*cos2x

Int sinx*cos 2x dx

To calculate the integral, we'll have to transform the product of trigonometric functions into an algebraic sum:

We'll use the product formula for:

sin a*cos b = [sin(a-b)+sin(a+b)]/2

We'll substitute a by x and b by 2x:

sinx*cos 2x = [sin(x-2x)+sin(x+2x)]/2

sinx*cos 2x = (sin3x - sin x)/2

We'll substitute the product by the difference:

Int sinx*cos 2x dx = Int (sin3x - sin x)dx/2

Int (sin3x - sin x)dx/2 = (1/2)*[Int (sin3x)dx - Int (sin x)dx]

Int (sin3x - sin x)dx/2 = -cos 3x/6 + cos x/2 + C