三角関数の積分法(integration of trigonometric functions)

確認問題

1.

次の積分を求めよう．

(a) $\displaystyle{\int{\sin^3{x}\cos{x}} dx}$ (b) $\displaystyle{\int{\sin^2{3x}\cos{3x}} dx}$ (c) $\displaystyle{\int{\cos^2{x}} dx}$

(d) $\displaystyle{\int{\cos^{3}{x}} dx }$ (e) $\displaystyle{\int{\cos^{4}{x}\sin^3{x}} dx}$ (f) $\displaystyle{\int{\sin{2x}\cos{3x}} dx}$

(g) $\displaystyle{\int{\sin{2x}\sin{x}} dx}$ (h) $\displaystyle{\int{\cos{x}\cos{2x}} dx}$ (i) $\displaystyle{\int{\tan{x}\sec^2{x}} dx}$

(j) $\displaystyle{\int{\sec^3{x}} dx}$

演習問題

1.

次の積分を求めよう．

(a) $\displaystyle{\int{\sin^3{x}} dx}$ (b) $\displaystyle{\int{\sin^2{3x}} dx}$ (c) $\displaystyle{\int{\sin^3{x}\cos^2{x}} dx}$

(d) $\displaystyle{\int{\cos{3x}\sin{2x}} dx }$ (e) $\displaystyle{\int{\sin^5{x}} dx}$ (f) $\displaystyle{\int{\sec^2{\pi x}} dx}$ (g) $\displaystyle{\int{\tan^3{x}} dx}$

(h) $\displaystyle{\int{\tan^2{x}\sec^2{x}} dx}$ (i) $\displaystyle{\int{\tan^3{x}\sec^3{x}} dx}$ (j) $\displaystyle{\int{\sec^5{x}} dx}$

(k) $\displaystyle{\int{\frac{dx}{3 - 2\cos{x}}}}$ (l) $\displaystyle{\int{\frac{\sin{x}}{2 - \sin{x}}} dx}$ (m) $\displaystyle{\int{\frac{1 + \sin{x}}{1 + \cos{x}}} dx}$

(n) $\displaystyle{\int{\frac{\sin^2{x}}{\sin^2{x} - \cos^2{x}}} dx}$ (o) $\displaystyle{\int{\frac{dx}{1 + \tan{x}}}}$