task/29641416
∫ cos2φ /(cosφ -sinφ) dφ =∫ (cos²φ-sin²φ) /(cosφ -sinφ) dφ =
=∫ (cosφ+sinφ)(cosφ-sinφ /(cosφ -sinφ) dφ =∫ (cosφ+sinφ)dφ =
= ∫cosφdφ +∫sinφdφ = sinφ -cosφ +C. ------
∫ 1 / ( tg(x/2) +ctg(x/2) ) dx =∫1 / ( sin(x/2) / cos(x/2) +cos(x/2) /sin(x/2) ) dx=
=∫sin(x/2) *cos(x/2) / ( sin²(x/2) + cos²(x/2) dx =∫(1/2)*sinx dx = - (1/2)cosx +C.
* * * sin²(x/2) + cos²(x/2) =1 и 2sinα*cosα =sin2α тождества * * * ------
∫ dy /(y -3)¹⁰ = ∫(y -3)⁻ ¹⁰ dy = (y -3)⁻ ⁹ / ( -9 ) +C = - 1 / 9(y - 3) ⁹ +C .