### Instruction: Evaluate the given integral.$$\displaystyle \int { \cfrac { 1 }{ 3+2\sin { x } +\cos { x }  }  } dx$$.### Response: This integral looks tricky, but I notice that the denominator is a linear combination of sine and cosine, which makes me think of a trigonometric identity.
If I recall correctly, there is an identity that says $$\cos { \left( x+\frac { \pi  }{ 4 }  \right)  } =\frac { 1 }{ \sqrt { 2 }  } \left( \cos { x } +\sin { x }  \right) $$
So maybe I can use that to simplify the denominator by rewriting it as a multiple of $\cos { \left( x+\frac { \pi  }{ 4 }  \right)  }$.
To do that, I need to find what constant $k$ satisfies $$3+2\sin { x } +\cos { x } = k \cos { \left( x+\frac { \pi  }{ 4 }  \right)  }$$
To find $k$, I can plug in a convenient value of $x$, such as $x=0$, and then solve for $k$.
When $x=0$, I get $$3+2\sin { 0 } +\cos { 0 } = k \cos { \left( 0+\frac { \pi  }{ 4 }  \right)  }$$
Simplifying, I get $$4=k\frac { \sqrt { 2 }  }{ 2 }$$
Solving for $k$, I get $$k=2\sqrt { 2 } $$
So now I can rewrite the integral as $$\displaystyle \int { \cfrac { 1 }{ 2\sqrt { 2 } \cos { \left( x+\frac { \pi  }{ 4 }  \right)  }  }  } dx$$
This looks much nicer, and I recognize that the antiderivative of $\cfrac { 1 }{ \cos { \left( x+\frac { \pi  }{ 4 }  \right)  }  } $ is $\tan { \left( x+\frac { \pi  }{ 4 }  \right)  } $, up to a constant.
So I can use the substitution rule to find the antiderivative.
Let $u=x+\frac { \pi  }{ 4 } $, then $du=dx$ and the integral becomes $$\displaystyle \int { \cfrac { 1 }{ 2\sqrt { 2 } \cos { u }  }  } du$$
The antiderivative is then $$\frac { 1 }{ 2\sqrt { 2 }  } \tan { u } +C$$
Substituting back for $u$, I get $$\frac { 1 }{ 2\sqrt { 2 }  } \tan { \left( x+\frac { \pi  }{ 4 }  \right)  } +C$$