[tex]\displaystyle\limit\int\sin(x+1)dx=\\
\text{Notam:}\\
x+1=t\\
(x+1)dx=dt\\
dx=dt\\
\displaystyle\limit\int \sin t\ dt=-\cos t+C\\
\text{Deci:}\displaystyle\int \sin(x+1)\ dx=-\cos(x+1)+C\\
\text{Prin urmare:}\\
\displaystyle\int_{0}^1 \sin (x+1)\ dx=-\cos 2+\cos 1=\boxed{\cos 1-\cos 2} [/tex]