Evaluate the following integrals:

We know cos²x + sin²x = 1.

Also, 2sinxcosx = sin2x

1 + sin2x =cos²x + sin²x + 2sinxcosx = (cosx + sinx)²

1 - sin2x =cos²x + sin²x - 2sinxcosx = (cosx - sinx)²

∴ The equation becomes

⇒

⇒

Assume cosx + sinx = t

∴ d(cosx + sinx) = dt

= cosx - sinx

∴ dt = cosx – sinx

⇒

= ln|t| + c

But t = cosx + sinx

∴ ln|cosx + sinx| + c.