Evaluate the following integrals:

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

∴ Denominator can be written as

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

∴ Now the given equation becomes

⇒

Assume cosx + sinx = t

∴ d(cosx + sinx) = dt

= cosx - sinx

∴ dt = cosx – sinx

⇒

⇒

⇒

⇒

But t = cosx + sinx

⇒