
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
Let the definite integral  $ I_n\equal{}\int_0^{\frac{\pi}{4}} \frac{dx}{(\cos x)^n}\ (n\equal{}0,\ \pm 1,\ \pm 2,\ \cdots )$ .

(1) Find  $ I_0,\ I_{\minus{}1},\ I_2$ .

(2) Find  $ I_1$ .

(3) Express  $ I_{n\plus{}2}$  in terms of  $ I_n$ .

(4) Find  $ I_{\minus{}3},\ I_{\minus{}2},\ I_3$ .

(5) Evaluate the definite integrals  $ \int_0^1 \sqrt{x^2\plus{}1}\ dx,\ \int_0^1 \frac{dx}{(x^2\plus{}1)^2}\ dx$  in using the avobe results.

You are not allowed to use the formula of integral for  $ \sqrt{x^2\plus{}1}$  directively here.

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