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.