
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Calculate the following indefinite integrals.

[1]  $\int \frac{x}{\sqrt{5-x}}dx$ 

[2]  $\int \frac{\sin x \cos ^2 x}{1+\cos x}dx$ 

[3]  $\int (\sin x+\cos x)^2dx$ 

[4]  $\int \frac{x-\cos ^2 x}{x\cos^ 2 x}dx$ 

[5] $\int (\sin x+\sin 2x)^2 dx$ 

Present the answer in LaTex format: \boxed{Your answer}
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