Question

im having a hard time trying to solve this question can you please solve with steps being explained

73. Find the indefinite integral, first by using an appropriate y-substitution, then using integration by parts: cos (Inr) dr (Hint: recall that I = e, if and only if, y = In ) E

Answer 1

Scos(lux) da Inx=y by parts]. is flos (tan) da Let =ffost - dy = – de y deux [x = 24 as y= ur]. Scos(lan, da = ((Cosyje e dy & Cosy set dy + s(siny se dy dy sa sed. Cosytet sinydy - 1) [ Applying integreation let, I set sinydy - (+). = siny Jeddy-f(cosy feddy) dy [integrating by - et siny- I cosy et dy parts = et siny - { cosy feldy + fliny jet dy dy} - et sing. et cosy- & siny. et dy. et (siny-Cosy)- I thi [by (+)]. : 2I = ey (siny- Cosy)+G. [C, is const.]. -) I = 1 (siny- Cosy) tc. [l is a const? f Cos (Lux) dx = et cosy + I by 21) & ()]

is the 1 et cosy + t ex siny-Cosy). - ey (Cosyt siny) + e, integrating const) (ucherec Scos (lan) de = I dual cos((un) + Sin (Lux))+c. - + x (cos (nm) + sin(x))+c
Here is one way of evaluating the integral