Question

Solve the following problems.

\( \cos (2 t+y) d y=d t \)

Answer 1

Solution

Solve the following problems.

\( \cos (2 t+y) d y=d t \)

Let \( u=2 t+y \Longrightarrow \frac{d u}{d t}=2+\frac{d y}{d t} \Longrightarrow \frac{d y}{d t}=\frac{d u}{d t}-2 \)

\( \Longrightarrow \cos (2 t+y) d y=d t \Longrightarrow \frac{d y}{d t}=\frac{1}{\cos (2 t+y)} \)

\( \Longrightarrow \frac{d u}{d t}=2+\frac{1}{\cos u} \)

\( \Longrightarrow \frac{d u}{d t}=\frac{2+\cos u}{\cos u} \Longrightarrow \frac{\cos u}{2+\cos u} d u=d t \)

\( \Longrightarrow \int \frac{\cos u}{2+\cos u} d u=\int d t \)

\( \Longrightarrow u-\tan \left(\frac{u}{2}\right)=t+C \)

 

 Therefore. \( (2 t+y)-\tan \left(\frac{2 t+y}{2}\right)=t+C \)