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

[2]  $\int x\log_{10} x dx$ 

[3]  $\int \frac{x}{\sqrt{2x-1}}dx$ 

[4]  $\int (x^2+1)\ln x dx$ 

[5]  $\int e^x\cos x dx$ 

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