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不定積分(7)

学籍番号
氏　　名
不定積分を行い、解を選択肢から選びなさい．
(A) \(\LARGE \int \normalsize \frac{\cos{\left(x \right)}}{\sqrt{\sin{\left(x \right)} + 2}} dx\) ()
(B) \(\LARGE \int \normalsize \left(1 - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} dx\) ()
(C) \(\LARGE \int \normalsize 3 \left(1 - \cos^{2}{\left(x \right)}\right)^{2} \sin{\left(x \right)} dx\) ()
(D) \(\LARGE \int \normalsize \sin^{5}{\left(x \right)} dx\) ()
(E) \(\LARGE \int \normalsize 1 - \cos{\left(2 x \right)} dx\) ()
(F) \(\LARGE \int \normalsize \sin^{2}{\left(x \right)} dx\) ()
(G) \(\LARGE \int \normalsize 8 \left(1 - \cos{\left(2 x \right)}\right)^{2} dx\) ()
(H) \(\LARGE \int \normalsize \sin^{4}{\left(x \right)} dx\) ()
(I) \(\LARGE \int \normalsize \frac{\tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} dx\) ()
(J) \(\LARGE \int \normalsize x e^{- x^{2}} dx\) ()
(K) \(\LARGE \int \normalsize \frac{\log{\left(x \right)}}{x \left(\log{\left(x \right)} + 1\right)^{2}} dx\) ()

関数に対数が含まれる場合，変数は対数の定義範囲とする．
\(\cos{\left(x + y \right)} = - \sin{\left(x \right)} \sin{\left(y \right)} + \cos{\left(x \right)} \cos{\left(y \right)}\)
\(\cos{\left(2 x \right)} = 2 \cos^{2}{\left(x \right)} - 1 = 1 - 2 \sin^{2}{\left(x \right)}\)
\(\cos{\left(3 x \right)} = 4 \cos^{3}{\left(x \right)} - 3 \cos{\left(x \right)}\)
\(\cos{\left(4 x \right)} = 2 \cos^{2}{\left(2 x \right)} - 1 = 1 - 2 \sin^{2}{\left(2 x \right)}\)
\(\cos{\left(5 x \right)} = 16 \cos^{5}{\left(x \right)} - 20 \cos^{3}{\left(x \right)} + 5 \cos{\left(x \right)}\)


選択肢

関数（\( C \) は積分定数）
(1)\(2 \sqrt{\sin{\left(x \right)} + 2} + C\) (2)\(\frac{3 x}{8} - \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(4 x \right)}}{32} + C\)
(3)\(16 \cos^{5}{\left(x \right)} - 20 \cos^{3}{\left(x \right)} + 5 \cos{\left(x \right)} + C\) (4)\(x - \frac{\sin{\left(2 x \right)}}{2} + C\)
(5)\(- \frac{\cos^{5}{\left(x \right)}}{5} + \frac{2 \cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)} + C\) (6)\(\frac{\tan^{3}{\left(x \right)}}{3} + C\)
(7)\(- \frac{e^{- x^{2}}}{2} + C\) (8)\(12 x - 8 \sin{\left(2 x \right)} + \sin{\left(4 x \right)} + C\)
(9)\(\frac{\cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)} + C\) (10)\(\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4} + C\)
(11)\(\log{\left(\log{\left(x \right)} + 1 \right)} + \frac{1}{\log{\left(x \right)} + 1} + C\) (12)\(- \frac{3 \cos^{5}{\left(x \right)}}{5} + 2 \cos^{3}{\left(x \right)} - 3 \cos{\left(x \right)} + C\)