解答题 （共 15 题 ），解答过程应写出必要的文字说明、证明过程或演算步骤
$\int_{-\frac{1}{2}}^{\frac{1}{2}} \frac{(\arcsin x)^2}{\sqrt{1-x^2}} d x$ ；
$\int_{\frac{1}{\sqrt{2}}}^1 \frac{\sqrt{1-x^2}}{x^2} d x$ ；
$\int_0^2 \frac{x d x}{\left(x^2-2 x+2\right)^2}$
证明定积分公式：
$$
\begin{gathered}
I_n=\int_0^{\frac{\pi}{2}} \sin ^n x d x\left(=\int_0^{\frac{\pi}{2}} \cos ^n x d x\right) \\
=\left\{\begin{array}{l}
\frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdots \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi}{2}, n \text { 为正偶数, } \\
\frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdots \frac{4}{5} \cdot \frac{2}{3}, n \text { 为大于 } 1 \text { 的正奇数. }
\end{array}\right.
\end{gathered}
$$
$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 4 \cos ^4 \theta d \theta$ ．
$\int_{-\sqrt{2}}^{\sqrt{2}} \sqrt{8-2 y^2} d y$;
$\int_0^a x^2 \sqrt{a^2-x^2} d x$;
$\int_1^{\sqrt{3}} \frac{d x}{x^2 \sqrt{1+x^2}}$;
$\int_1^4 \frac{d x}{1+\sqrt{x}}$;
$\int_0^1 t e^{-\frac{t^2}{2}} d t$;
$\int_1^{e^2} \frac{d x}{x \sqrt{1+\ln x}}$;
$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sqrt{\cos x-\cos ^3 x} d x$;
$\int_0^\pi \sqrt{1+\cos 2 x} d x$.
证明 $\int_x^1 \frac{d x}{1+x^2}=\int_1^{\frac{1}{x}} \frac{d x}{1+x^2}(x>0)$.