Evaluate the integral shown below using the given method

Question

∫sin(x)cos(x) dx

 

1. substitution where u=sin(x)

2. substitution where u=cos(x)

3 .integration by parts

4. using the identity sin(2x)=2sin(x)cos(x)

Answer 1

1) u = sin x

du = cos x dx

 

Substitute

 

Int( u du) = (u^2)/2 + c = .5(sin x)^2+ c

 

2) u = cos x

 

du = - sin x dx

 

Int(-u du) = (-u^2)/2 + c = -.5(cos x)^2 + c

 

3) u = sin x

 

du = cos x dx

 

dv = cos x dx

 

v = sin x

 

Int( u*dv) = uv - Int(v*du)

 

Int(sin x * cos x) = (sin x)^2 - Int(sin x * cos x)

 

2*Int(sin x * cos x) = (sin x)^2

 

Int(sin x * cos x) = .5(sin x)^2 + c

 

4) Int(sin x * cos x) = Int(.5sin 2x) = -.25cos 2x + c

Answer 2

Hello Muhammad,

 

I feel, the constant of integrations will be different in all the four cases.

 

Surendra