置換積分法(integration by substitution)

確認問題

1.

次の積分を求めよう．

(a) $\displaystyle{\int{\sin{2x}}\ dx}$ (b) $\displaystyle{\int{\frac{x}{x^{2}+1}}\ dx}$ (c) $\displaystyle{\int{e^{2x}}\ dx}$ (d) $\displaystyle{\int{\frac{1}{x\log{x}}}\ dx}$

(e) $\displaystyle{\int{xe^{x^{2}}}\ dx}$ (f) $\displaystyle{\int{\sin^{2}{x}\cos{x}}\ dx}$ (g) $\displaystyle{\int{x\sqrt{1+x}} \ dx}$

(h) $\displaystyle{\int{\frac{\cos{x} - \sin{x}}{\sin{x} + \cos{x}}}\ dx }$ (i) $\displaystyle{\int{\frac{e^x}{1 + e^{x}}} \ dx}$

演習問題

1.: 次の積分を求めよう．
(a) $\displaystyle{\int{e^{2-x}}\ dx}$ (b) $\displaystyle{\int{\sec^2{(1-x)}}\ dx}$ (c) $\displaystyle{\int{\frac{x}{\sqrt{1-x^2}}}\ dx}$ (d) $\displaystyle{\int{\frac{\sin{x}}{\cos^2{x}}}\ dx}$ (e) $\displaystyle{\int{\frac{e^{1/x}}{x^2}} \ dx}$ (f) $\displaystyle{\int{\frac{\sec^2{\theta}}{\sqrt{3\tan{\theta} + 1}}}\ d\theta}$ (g) $\displaystyle{\int{\frac{1+\cos{2x}}{\sin^2{x}}} \ dx}$ (h) $\displaystyle{\int{\frac{\log{x}}{x}}\ dx }$ (i) $\displaystyle{\int{\frac{e^x}{1 + e^{2x}}} \ dx}$ (j) $\displaystyle{\int{x\sin{(x^2)}} \ dx}$