Integral of sin 2x divided by 1 plus cosine squared x?

Answer 1

3

Answer 2

Integral of [sin 2x/(1 + cos2 x)] dx (substitute 2sinx cos x for sin 2x)

= Integral of [2sin x cos x/(1 + cos2 x)] dx

Let u = 1 + cos2 x, so that du = -2cos x sinx dx, and dx = du/-2sin x cos x.

(Substitute u for 1 + cos x, and -2sin x cos x for dx). So we have:

= Integral of [(2sin x cos x/u)(du/-2sin x cos x)] (simplify)

= Integral of -du/u

= -ln |u| + C (substitute 1 + cos2 x for u, and take off the absolute value symbol since the expression inside it is always positive)

= -ln (1 + cos2 x) + C