6.6 演習～不定積分，置換積分，部分積分

問 6.27 (不定積分) 次の不定積分を書け．
    (1)   $\displaystyle{\int\,dx}$     (2)   $\displaystyle{\int x^3\,dx}$     (3)   $\displaystyle{\int x^4\,dx}$     (4)   $\displaystyle{\int\frac{dx}{x}}$     (5)   $\displaystyle{\int\frac{dx}{x^4}}$     (6)   $\displaystyle{\int\frac{dx}{x^5}}$     (7)   $\displaystyle{\int\sqrt{x}\,dx}$
    (8)   $\displaystyle{\int e^{x}\,dx}$     (9)   $\displaystyle{\int 2^{x}\,dx}$     (10)   $\displaystyle{\int 3^{x}\,dx}$     (11)   $\displaystyle{\int\sin x\,dx}$     (12)   $\displaystyle{\int\cos x\,dx}$     (13)   $\displaystyle{\int\frac{dx}{\cos^2x}}$
    (14)   $\displaystyle{\int\sinh x\,dx}$     (15)   $\displaystyle{\int\cosh x\,dx}$     (16)   $\displaystyle{\int\frac{dx}{\cosh^2x}}$     (17)   $\displaystyle{\int\frac{dx}{\sqrt{1-x^2}}}$     (18)   $\displaystyle{\int\frac{dx}{\sqrt{1+x^2}}}$
    (19)   $\displaystyle{\int\frac{dx}{\sqrt{x^2-1}}}$     (20)   $\displaystyle{\int\frac{dx}{1+x^2}}$     (21)   $\displaystyle{\int\frac{dx}{x^2-1}}$     (22)   $\displaystyle{\int\frac{dx}{1-x^2}}$

問 6.28 (不定積分の計算) 次の不定積分を求めよ．
    (1)   $\displaystyle{\int(5x^4+6x^2)\,dx}$     (2)   $\displaystyle{\int(x^2-3x+1)\,dx}$     (3)   $\displaystyle{\int(x^3-2x)\,dx}$     (4)   $\displaystyle{\int\left(x^7+\frac{1}{x^9}\right)\,dx}$
    (5)   $\displaystyle{\int(1+x)\sqrt[3]{x}\,dx}$     (6)   $\displaystyle{\int\sqrt[3]{1+x}\,\,dx}$     (7)   $\displaystyle{\int\frac{2x+3}{\sqrt x}\,dx}$     (8)   $\displaystyle{\int\frac{(x+1)^2}{x^2}\,dx}$     (9)   $\displaystyle{\int\frac{1+x^3}{x^4}\,dx}$
    (10)   $\displaystyle{\int\frac{1+x}{x^3}\,dx}$     (11)   $\displaystyle{\int\frac{1}{\sqrt[3]{x}}\,\,dx}$     (12)   $\displaystyle{\int \cos^2 x\,dx}$     (13)   $\displaystyle{\int(2\cos x - 3\sin x)\,dx}$
    (14)   $\displaystyle{\int(4\,\sin x-3\,\cos x)\,dx}$

問 6.29 (置換積分法) 次の不定積分を求めよ．
    (1)   $\displaystyle{\int 3x^2(x^3+2)^{3}\,dx}$     (2)   $\displaystyle{\int x^2(x^3+1)^{4}\,dx}$     (3)   $\displaystyle{\int(2x+5)^6\,dx}$     (4)   $\displaystyle{\int (2x^3+1)(x^4+2x)^{10}\,dx}$
    (5)   $\displaystyle{\int\frac{2x-1}{x^2-2x+1}\,dx}$     (6)   $\displaystyle{\int\frac{x}{(1+x^2)^3}\,dx}$     (7)   $\displaystyle{\int\frac{dx}{x^2+4}}$     (8)   $\displaystyle{\int\frac{x-1}{x^2-2x+1}\,dx}$     (9)   $\displaystyle{\int\frac{dx}{\sqrt{1+3x}}}$
    (10)   $\displaystyle{\int\sqrt{x^2-x^4}\,dx}$     (11)   $\displaystyle{\int\frac{1}{\sqrt{2x-x^2}}\,dx}$     (12)   $\displaystyle{\int\frac{x^2}{\sqrt{1-x^6}}\,dx}$     (13)   $\displaystyle{\int\frac{x}{\sqrt{1-(x+2)^2}}\,dx}$
    (14)   $\displaystyle{\int\frac{x^2}{\sqrt{2x^3-5}}\,dx}$     (15)   $\displaystyle{\int\frac{\cos x}{1+\sin x}\,dx}$     (16)   $\displaystyle{\int x\,\cos(2x^2+1)\,dx}$     (17)   $\displaystyle{\int\tanh x}\,\,dx$
    (18)   $\displaystyle{\int\frac{2\cos x}{1+3\sin x}\,dx}$     (19)   $\displaystyle{\int\frac{\sin x}{\cos^3 x}\,dx}$     (20)   $\displaystyle{\int\frac{dx}{\tan x}}$     (21)   $\displaystyle{\int xe^{x^2}\,\,dx}$     (22)   $\displaystyle{\int {\frac{\log x}{x}}\,dx}$
    (23)   $\displaystyle{\int\frac{dx}{e^{x}+e^{-x}}}$     (24)   $\displaystyle{\int\frac{dx}{x\,\log x}}$     (25)   $\displaystyle{\int(e^x+1)^2 e^x\,\,dx}$     (26)   $\displaystyle{\int\log x^{\frac{1}{x}}\,dx}$

問 6.30 (部分積分法) 次の不定積分を求めよ．
(1) $\displaystyle{\int x^2e^{x}\,dx}$ (2) $\displaystyle{\int x^2e^{-x}\,dx}$ (3) $\displaystyle{\int x\,\log x\,dx}$ (4) $\displaystyle{\int x^2\sin x\,dx}$ (5) $\displaystyle{\int x^2\cos x\,dx}$
(6) $\displaystyle{\int\mathrm{Sin}^{-1}x\,dx}$ (7) $\displaystyle{\int\mathrm{Tan}^{-1}x\,dx}$ (8) $\displaystyle{\int x\sqrt{1+x}\,dx}$

Kondo Koichi
平成19年1月23日

6.6 演習 ～ 不定積分，置換積分，部分積分

6.6 演習～不定積分，置換積分，部分積分