微積２中間：公式テスト

2023/12/07 修正

\(\log x = \log_e x\)である．111点満点．正答におけるCは積分定数とする．問題をタップで正答表示．

ネットに接続されていないと，数式が正しく表示されません！

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1 次の計算をせよ．（各１点）

(1) \((x^n)'\) \(nx^{n-1}\)
(2) \((x)'\) 1
(3) \((C)'\) 0
(4) \((e^x)'\) \(e^x\)
(5) \((a^x)'\) \(a^x \space \log a\)
(6) \(\left\{e^{f(x)}\right\}'\) \(f'e^f\)
(7) \((\log x)'\) \(\displaystyle\frac 1 x \)
(8) \((\log_a x)'\) \(\displaystyle\frac 1 {x \log a}\)
(9) \(\{\log {f(x)} \}'\) \(\displaystyle\frac {f'} f\)
(10) \((\sin x)'\) \(\cos x\)
(11) \(\{\sin {f(x)}\}'\) \(f' \cos f\)
(12) \((\cos x)'\) \(- \sin x\)
(13) \(\{\cos {f(x)}\}'\) \(-f' \sin f\)
(14) \((\tan x)'\) \(\displaystyle\frac 1 {\cos^2 x}\)
(15) \(\{\tan {f(x)}\}'\) \(\displaystyle\frac {f'} {\cos^2 f} \)
(16) \(\left\{ \displaystyle\frac 1 {\tan x} \right\}'\) \(\displaystyle\frac {-1} {\sin^2 x} \)
(17) \(\left\{ \displaystyle\frac 1 {\tan {f(x)}} \right\}'\) \(\displaystyle\frac {-f'} {\sin^2 f} \)
(18) \((\sin^{-1} x)'\) \(\displaystyle\frac {1} {\sqrt{1-x^2}} \)
(19) \(\{\sin^{-1} {f(x)}\}'\) \(\displaystyle\frac {f'} {\sqrt{1-f^2}} \)
(20) \((\tan^{-1} x)'\) \(\displaystyle\frac {1} {1+x^2} \)
(21) \(\{\tan^{-1} {f(x)}\}'\) \(\displaystyle\frac {f'} {1+f^2} \)
(22) \(F' (x)\) \(f\)
(23) \(F'(ax+b)\) \(af(ax+b)\) \(\displaystyle\frac {dy} {dx} = \frac {dy} {du} \cdot \frac {du} {dx} \) （合成関数の微分法）

2 次の計算をせよ．（各１点）

(1) \(\displaystyle\int x^n \space dx\) \(\displaystyle\frac {1} {n+1} x^{n+1} +C\)
(2) \(\displaystyle\int 1 \space dx\) \(x+C\)
(3) \(\displaystyle\int 0 \space dx\) \(C\)
(4) \(\displaystyle\int e^x \space dx\) \(e^x +C\)
(5) \(\displaystyle\int a^x \space dx\) \(\displaystyle\frac {1} {\log a} a^x +C\)
(6) \(\displaystyle\int e^{ax+b} \space dx\) \(\displaystyle\frac {1} {a} e^{ax+b} +C\)
(7) \(\displaystyle\int \frac 1 x \space dx\) \(\log {|x|} +C\)
(8) \(\displaystyle\int \frac 1 {ax+b} \space dx\) \(\displaystyle\frac {1} {a} \log {|ax+b|} +C\)
(9) \(\displaystyle\int \cos x \space dx\) \(\sin x +C\)
(10) \(\displaystyle\int \cos{(ax+b)} \space dx\) \(\displaystyle\frac {1} {a} \sin {(ax+b)} +C\)
(11) \(\displaystyle\int \sin x \space dx\) \(-\cos x +C\)
(12) \(\displaystyle\int \sin{(ax+b)} \space dx\) \(-\displaystyle\frac {1} {a} \cos {(ax+b)} +C\)
(13) \(\displaystyle\int \frac 1 {\cos^2 x} \space dx\) \(\tan x +C\)
(14) \(\displaystyle\int \frac 1 {\cos^2{(ax+b)}} \space dx\) \(\displaystyle\frac {1} {a} tan {(ax+b)} +C\)
(15) \(\displaystyle\int \frac 1 {\sin^2 x} \space dx\) \(-\displaystyle\frac {1} {\tan x} +C\)
(16) \(\displaystyle\int \frac 1 {\sin^2 {(ax+b)}} \space dx\) \(-\displaystyle\frac {1} {a\tan {(ax+b)}} +C\)
(17) \(\displaystyle\int \frac 1 {\sqrt{1-x^2}} \space dx\) \(\sin^{-1} x +C\)
(18) \(\displaystyle\int \frac 1 {\sqrt{a^2-x^2}} \space dx\) \(\displaystyle \sin^{-1} {\frac x a} +C\)
(19) \(\displaystyle\int \frac 1 {\sqrt{1-a^2x^2}} \space dx\) \(\displaystyle\frac {1} {a} \sin^{-1} {(ax)} +C\)
(20) \(\displaystyle\int \frac 1 {1+x^2} \space dx\) \(\tan^{-1} x +C\)
(21) \(\displaystyle\int \frac 1 {a^2+x^2} \space dx\) \(\displaystyle\frac {1} {a} \tan^{-1} {\frac x a} +C\)
(22) \(\displaystyle\int \frac 1 {1+a^2x^2} \space dx\) \(\displaystyle\frac {1} {a} tan^{-1} {(ax)} +C\)
(23) \(\displaystyle\int f(ax+b) \space dx\) \(\displaystyle\frac {1} {a} F(ax+b) +C\) \(\displaystyle\frac {dy} {dx} = \frac {dy} {du} \cdot \frac {du} {dx} \) （合成関数の微分法）

以下，(24)~(26)は公式テスト範囲外．
しかし試験範囲ではある．

(24) \(\displaystyle\int \frac {1} {\sqrt{k^2-(px+q)^2}} \space dx\) \(\displaystyle \frac{1}{p} \sin^{-1} \frac{px+q}{k} + C \) ( \(k=0\) )
(25) \(\displaystyle\int \frac {1} {k^2+(px+q)^2} \space dx\) \(\displaystyle \frac{1}{pk} \tan^{-1} \frac{px+q}{k} + C \) ( \(k=0\) )
(26) \(\displaystyle\int \frac{1}{x^2-a^2} \space dx\) \(\displaystyle \frac{1}{2a} \log { \frac{|x-a|}{|x+a|} } + C \)

3 次の計算をせよ．（各２点）

(1) \((x^{11})'\) \(11x^{10}\)
(2) \((x)'\) \(1\)
(3) \((1)'\) \(0\)
(4) \((e^{2x})'\) \(2e^{2x}\)
(5) \((3^x)'\) \(3^x \log 3\)
(6) \(\left\{\log {(4x+5)} \right\}'\) \(\displaystyle\frac 4 {4x+5}\)
(7) \((\log_6 x)'\) \(\displaystyle\frac 1 {x \log 6}\)
(8) \((\sin {7x})'\) \(7 \cos {7x}\)
(9) \((\cos {8x})'\) \(-8 \sin {8x}\)
(10) \((\tan {9x})'\) \(\displaystyle\frac 9 {\cos^2 {9x}}\)
(11) \(\left\{kf(x)+lg(x)\right\}'\) \(kf'+lg'\)
(12) \(\left\{f(x)g(x)\right\}'\) \(f'g+fg'\)
(13) \(\left\{ \displaystyle\frac {f(x)} {g(x)} \right\}'\) \(\displaystyle\frac {f'g-fg'} {g^2}\)
(14) \(\left\{ \displaystyle\frac 1 {g(x)} \right\}'\) \(-\displaystyle\frac {g'} {g^2}\)
(15) \(\left\{ \displaystyle\frac 1 {\tan {2x}} \right\}'\) \(-\displaystyle\frac 2 {\sin^2 {2x}}\)
(16) \((\sin^{-1} 3x)'\) \(\displaystyle\frac 3 {\sqrt{1-9x^2}}\)
(17) \((\tan^{-1} 4x)'\) \(\displaystyle\frac 4 {1+16x^2}\)

4 次の計算をせよ．（各２点）

(1) \(\displaystyle\int x^{10} \space dx\) \(\displaystyle\frac 1 {11} x^{11} + C\)
(2) \(\displaystyle\int 1 \space dx\) \(x +C\)
(3) \(\displaystyle\int 0 \space dx\) \(C\)
(4) \(\displaystyle\int e^{2x} \space dx\) \(\displaystyle\frac 1 2 e^{2x} +C\)
(5) \(\displaystyle\int 3^x \space dx\) \(\displaystyle\frac 1 {\log 3} 3^x +C\)
(6) \(\displaystyle\int \frac 1 {4x+5} \space dx\) \(\displaystyle\frac 1 4 \log {|4x+5|} +C\)
(7) \(\displaystyle\int \cos 7x \space dx\) \(\displaystyle\frac 1 7 \sin {7x} +C\)
(8) \(\displaystyle\int \sin 8x \space dx\) \(-\displaystyle\frac 1 8 \cos {8x} +C\)
(9) \(\displaystyle\int \frac 1 {\cos^2 9x} \space dx\) \(\displaystyle\frac 1 9 \tan {9x} +C\)
(10) \(\displaystyle\int \frac 1 {\sin^2 2x} \space dx\) \(-\displaystyle\frac 1 {2 \tan {2x}} +C\)
(11) \(\displaystyle\int \frac 1 {\sqrt{1-9x^2}} \space dx\) \(\displaystyle\frac 1 3 \sin^{-1} 3x +C\)
(12) \(\displaystyle\int \frac 1 {\sqrt{9-x^2}} \space dx\) \(\displaystyle\sin^{-1} {\frac x 3} +C\)
(13) \(\displaystyle\int \frac 1 {1+16x^2} \space dx\) \(\displaystyle\frac 1 4 \tan^{-1} 4x +C\)
(14) \(\displaystyle\int \frac 1 {16+x^2} \space dx\) \(\displaystyle\frac 1 4 \tan^{-1} {\frac x 4} +C\)