Integralni bajarishdan oldin biz mahsulotni ko'paytirishimiz kerak va kosekantning sinus bo'yicha ta'rifini unutmang.

\[\begin{align*}\int{{12 + \csc \left( \theta \right)\left[ {\sin \left( \theta \right) + \csc \left( \theta \right)} \right]\,d\theta }} & = \int{{12 + \csc \left( \theta \right)\sin \left( \theta \right) + {{\csc }^2}\left( \theta \right)\,d\theta }}\\ & = \int{{13 + {{\csc }^2}\left( \theta \right)\,d\theta }}\end{align*}\]

Eslatib o'tamiz,

\[\csc \left( \theta \right) = \frac{1}{{\sin \left( \theta \right)}}\]

va shuning uchun,

\[\csc \left( \theta \right)\sin \left( \theta \right) = 1\]

Buni qilish bizga integralni ancha soddalashtirishga imkon beradi va aslida integralni amalga oshirishga imkon beradi. Ushbu soddalashtirishsiz biz ikkinchi atamani hozirda mavjud bo'lgan bilimlar bilan birlashtira olmas edik.

2-qadam

Hozirgi vaqtda integralni baholashdan boshqa hech narsa qilish kerak emas.

\[\int{{12 + \csc \left( \theta \right)\left[ {\sin \left( \theta \right) + \csc \left( \theta \right)} \right]\,d\theta }} = \int{{13 + {{\csc }^2}\left( \theta \right)\,d\theta }} = \require{bbox} \bbox[2pt,border:1px solid black]{{13\theta - \cot \left( \theta \right) + c}}\]

Shuni unutmangki, trig funktsiyalari bilan ba'zi atamalar trig funktsiyalarining ta'rifini va/yoki ularning boshqa trig funktsiyalari bilan aloqasini eslash orqali sezilarli darajada soddalashtirilishi mumkin.