Evaluate the following integrals:

Expanding (cos x + sin x)² = cos²x + sin²x + 2 sinx cosx

We know cos²x + sin²x = 1, 2sinxcosx = sin2x

∴ (cos x + sin x)² = 1 + sin2x

∴ we can write the given equation as

⇒

Assume 1 + sin2x = t

⇒

⇒ 2cos2x dx = dt

∴ cos2xdx =

Substituting these values in the above equation we get

⇒

⇒ + c

substituting t = 1 + 2 sin x in above equation

⇒