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 .

Verified answer

I hope this helps you