$$ I = \int \frac { \cos x } { \sqrt { 4 + \sin x } } d $$

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Calculate or simplify the expression $$ \int \frac{\cos(x)}{\sqrt{4+\sin(x)}}dx $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int \frac{\cos\left(xight)}{\sqrt{4+\sin\left(xight)}} dx$$
- step1: Rewrite the expression:
  $$\int \frac{\cos\left(xight)}{\left(4+\sin\left(xight)ight)^{\frac{1}{2}}} dx$$
- step2: Use the substitution $$dx=\frac{1}{\cos\left(xight)} dt$$ to transform the integral$$:$$
  $$\int \frac{\cos\left(xight)}{\left(4+\sin\left(xight)ight)^{\frac{1}{2}}}	imes \frac{1}{\cos\left(xight)} dt$$
- step3: Simplify:
  $$\int \frac{1}{\left(4+\sin\left(xight)ight)^{\frac{1}{2}}} dt$$
- step4: Use the substitution $$t=\sin\left(xight)$$ to transform the integral$$:$$
  $$\int \frac{1}{\left(4+tight)^{\frac{1}{2}}} dt$$
- step5: Use the substitution $$dt=1 dv$$ to transform the integral$$:$$
  $$\int \frac{1}{\left(4+tight)^{\frac{1}{2}}}	imes 1 dv$$
- step6: Simplify:
  $$\int \frac{1}{\left(4+tight)^{\frac{1}{2}}} dv$$
- step7: Use the substitution $$v=4+t$$ to transform the integral$$:$$
  $$\int \frac{1}{v^{\frac{1}{2}}} dv$$
- step8: Evaluate the integral:
  $$\frac{v^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}$$
- step9: Add the numbers:
  $$\frac{v^{\frac{1}{2}}}{-\frac{1}{2}+1}$$
- step10: Add the numbers:
  $$\frac{v^{\frac{1}{2}}}{\frac{1}{2}}$$
- step11: Substitute back:
  $$\frac{\left(4+tight)^{\frac{1}{2}}}{\frac{1}{2}}$$
- step12: Substitute back:
  $$\frac{\left(4+\sin\left(xight)ight)^{\frac{1}{2}}}{\frac{1}{2}}$$
- step13: Rewrite the expression:
  $$2\left(4+\sin\left(xight)ight)^{\frac{1}{2}}$$
- step14: Simplify:
  $$2\sqrt{4+\sin\left(xight)}$$
- step15: Add the constant of integral C:
  $$2\sqrt{4+\sin\left(xight)} + C, C \in \mathbb{R}$$
The integral of $$ \frac{\cos(x)}{\sqrt{4+\sin(x)}} $$ with respect to $$ x $$ is $$ 2\sqrt{4+\sin(x)} + C $$, where $$ C $$ is an arbitrary constant.
The integral of $$ \frac{\cos(x)}{\sqrt{4+\sin(x)}} $$ with respect to $$ x $$ is $$ 2\sqrt{4+\sin(x)} + C $$, where $$ C $$ is a constant.

\( I = \int \frac { \cos x } { \sqrt { 4 + \sin x } } d \)

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