Question

cos x sin x14.1sin 2x

Answer 1

The given function is cosx−sinx 1+sin2x .

∫ cosx−sinx 1+sin2x dx (1)

Also, sin 2 x+ cos 2 x=1 and sin2x=2sinxcosx (2)

From (1) and (2),

cosx−sinx 1+sin2x = cosx−sinx ( sinx+cosx ) 2

Let,

sinx+cosx=t ( cosx−sinx )dx=dt (3)

Put (3) in (1), we get,

∫ cosx−sinx 1+sin2x dx = ∫ dt t 2 =− 1 t +c (4)

Substitute t in (4), we get

∫ cosx−sinx 1+sin2x dx = −1 sinx+cosx +c

Thus, the integral of the function cosx−sinx 1+sin2x is −1 sinx+cosx +c.