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

演習問題

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