定積分の計算(calculation of integrals)

演習問題

1.: 次の定積分の値を求めよう．
(a): $\displaystyle{\int_{0}^{\frac{\pi}{2}}\cos^{4}{x}\sin{x}dx}$
(b): $\displaystyle{\int_{0}^{\frac{\pi}{2}}\frac{\sin{x}}{\sqrt{1+\cos{x}}} dx}$
(c): $\displaystyle{\int_{0}^{\frac{\pi}{2}}\sin^{4}{x}dx}$
(d): $\displaystyle{\int_{-1}^{1}x^2 \cos{x} dx}$
(e): $\displaystyle{\int_{0}^{\pi}\cos{nx}dx}$ (n 整数)
2.: $\displaystyle{\int_{0}^{\frac{\pi}{2}}\sin^{n}{x}dx = \int_{0}^{\frac{\pi}{2}}\cos^{n}{x}dx}$ を示そう．