無理関数の積分法(integration of irrational functions)

確認問題

1.

次の積分を求めよう．

(a) $\displaystyle{\int{\frac{1}{1+\sqrt{x}}}\ dx}$ (b) $\displaystyle{\int{\frac{\sqrt{x}}{\sqrt{x} + 1}}\ dx}$ (c) $\displaystyle{\int{\frac{dx}{\sqrt{1 - e^x}}}}$ (d) $\displaystyle{\int{\frac{x}{\sqrt{x - 1}}}\ dx}$ (e) $\displaystyle{\int{\frac{x}{\sqrt{x^2 + 4}}}\ dx}$ (f) $\displaystyle{\int{\frac{x^{3}}{\sqrt{x^2 + 4}}}\ dx}$ (g) $\displaystyle{\int{\frac{dx}{\sqrt{x^2 - 2x - 3}}}}$

(h) $\displaystyle{\int{\frac{\sqrt{x^{2}-1}}{x}}\ dx}$

演習問題

1.

次の積分を求めよう．

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

(e) $\displaystyle{\int{\sqrt{\frac{x+1}{x-1}}}\ dx}$

2.

次の積分を求めよう．

(a) $\displaystyle{\int{\frac{x}{\sqrt{x^2 - 4}}}\ dx}$ (b) $\displaystyle{\int{\frac{x^2}{\sqrt{4 - x^2}}}\ dx}$ (c) $\displaystyle{\int{\frac{e^x}{9 - e^{2x}}}\ dx}$ (d) $\displaystyle{\int{\frac{\sqrt{1 - x^2}}{x^4}}\ dx}$ (e) $\displaystyle{\int{\frac{dx}{x^2\sqrt{x^2 - a^2}}}}$ (f) $\displaystyle{\int{\frac{dx}{e^x\sqrt{4 + e^{2x}}}}}$ (g) $\displaystyle{\int{\frac{dx}{\sqrt{x^2 - 2x - 3}}}}$

(h) $\displaystyle{\int{\frac{x}{\sqrt{6x - x^2}}}\ dx}$ (i) $\displaystyle{\int{\frac{x}{\sqrt{x^2 - 2x - 3}}}\ dx}$ (j) $\displaystyle{\int{\sqrt{6x - x^2 - 8}}\ dx}$

(k) $\displaystyle{\int{x\sqrt{x^2 + 6x}}\ dx}$