Question

10.1-2sin2 xcos'x

Answer 1

The given function f( x ) is sin 8 x− cos 8 x 1−2 sin 2 x cos 2 x .

Integrate both sides,

∫ f( x )dx = ∫ sin 8 x− cos 8 x 1−2 sin 2 x cos 2 x dx = ∫ ( sin 4 x ) 2 − ( cos 4 x ) 2 1−2 sin 2 x cos 2 x dx

On solving further we get,

∫ f( x )dx = ∫ ( sin 4 x− cos 4 x )( sin 4 x+ cos 4 x ) 1−2 sin 2 x cos 2 x dx = ∫ { ( ( sin 2 x ) 2 − ( cos 2 x ) 2 ) }{ ( ( sin 2 x ) 2 + ( cos 2 x ) 2 ) } 1−2 sin 2 x cos 2 x dx = ∫ ( sin 2 x− cos 2 x )( sin 2 x+ cos 2 x ){ ( sin 2 x+ cos 2 x ) 2 −2 sin 2 x cos 2 x } 1−2 sin 2 x cos 2 x dx

Further simplify.

∫ f( x )dx = ∫ ( sin 2 x− cos 2 x )( 1−2 sin 2 x cos 2 x ) 1−2 sin 2 x cos 2 x dx = ∫ ( sin 2 x− cos 2 x )dx = ∫ −cos2xdx =− sin2x 2 +c

Thus, integration of f( x )is − sin2x 2 +c.