微積公式集
2024/03/26
\(\log x = \log_e x\)である．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／非表示．
どこかしらに間違いや不足等あると思われます（特に変域とかは曖昧）．見つけたら是非ご連絡ください．
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0 基礎 多分半分以上書いてない
＜四則演算の微分法＞
\( (kf(x) \pm lg(x))' = kf'(x) \pm lg'(x) \)
\( \begin{eqnarray} \displaystyle \left \{f(x)g(x) \right \}' &=& f'(x)g(x) + f(x)g'(x) \\ & \neq & f'(x)g'(x) \end{eqnarray} \)
\( \begin{eqnarray} \displaystyle \left \{ \frac{f(x)}{g(x)} \right \}' &=& \frac{f'(x)g(x) - f(x)g'(x)}{g^2} \\ & \neq & \frac{f'(x)}{g'(x)} \end{eqnarray} \)
＜合成関数の微分法＞
＜逆関数の微分法＞
1 微分の公式
(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) \( \displaystyle \left(\sin^{-1}{\frac{x}{a}}\right)' \)
\( \displaystyle \frac{1}{\sqrt{a^2-x^2}} \)
(21) \((\cos^{-1} x)'\)
\(\displaystyle - \frac {1} {\sqrt{1-x^2}} \)
(22) \(\{\cos^{-1} {f(x)}\}'\)
\(\displaystyle - \frac {f'} {\sqrt{1-f^2}} \)
(23) \((\tan^{-1} x)'\)
\(\displaystyle\frac {1} {1+x^2} \)
(24) \(\{\tan^{-1} {f(x)}\}'\)
\(\displaystyle\frac {f'} {1+f^2} \)
(25) \( \displaystyle \left(\tan^{-1}{\frac{x}{a}}\right)' \)
\( \displaystyle \frac{a}{a^2+x^2} \)
(26) \(F' (x)\)
\(f\)
(27) \(F'(ax+b)\)
\(af(ax+b)\)
\(\displaystyle\frac {dy} {dx} = \frac {dy} {du} \cdot \frac {du} {dx} \)
（合成関数の微分法）
2 不定積分の公式
(1) \(\displaystyle\int 0 \space dx\)
\(C\)
(2) \(\displaystyle\int 1 \space dx\)
\(x+C\)
(3) \(\displaystyle\int x^n \space dx\)
\(\displaystyle\frac {1} {n+1} x^{n+1} +C \ (n \neq -1)\)
(3') \(\displaystyle\int (px+q)^n \space dx\)
\(\displaystyle\frac {1} {p(n+1)} (px+q)^{n+1} +C \ (n \neq -1,\) \( 以下，p \neq 0 \))
(5) \(\displaystyle\int e^x \space dx\)
\(e^x +C\)
(4) \(\displaystyle\int \frac 1 x \space dx\)
\(\log {\left | x \right |} +C \ (x \neq 0)\)
(4') \(\displaystyle\int \frac 1 {px+q} \space dx\)
\(\displaystyle\frac {1} {p} \log {\left | px+q \right |} +C\)
(5') \(\displaystyle\int e^{px+q} \space dx\)
\(\displaystyle\frac {1} {p} e^{px+q} +C \)
(6) \(\displaystyle\int a^x \space dx\)
\(\displaystyle\frac {1} {\log a} a^x +C \ (a>0, a \neq 1) \)
(6') \(\displaystyle\int a^{px+q} \space dx\)
\(\displaystyle\frac {1} {p \log a} a^{px+q} +C \ (a>0, a \neq 1) \)
(7) \(\displaystyle\int \sin x \space dx\)
\(-\cos x +C\)
(7') \(\displaystyle\int \sin{(px+q)} \space dx\)
\(-\displaystyle\frac {1} {p} \cos {(px+q)} +C\)
(8) \(\displaystyle\int \cos x \space dx\)
\(\sin x +C\)
(8') \(\displaystyle\int \cos{(px+q)} \space dx\)
\(\displaystyle\frac {1} {p} \sin {(px+q)} +C \)
(9) \(\displaystyle\int \frac 1 {\cos^2 x} \space dx\)
\(\tan x +C\)
(9') \(\displaystyle\int \frac 1 {\cos^2{(px+q)}} \space dx\)
\(\displaystyle\frac {1} {p} tan {(px+q)} +C\)
(10) \(\displaystyle\int \frac 1 {\sin^2 x} \space dx\)
\(-\displaystyle\frac {1} {\tan x} +C\)
(10') \(\displaystyle\int \frac 1 {\sin^2 {(px+q)}} \space dx\)
\(-\displaystyle\frac {1} {p\tan {(px+q)}} +C\)
(11) \(\displaystyle\int \frac 1 {\sqrt{1-x^2}} \space dx\)
\(\sin^{-1} x +C\)
(11') \(\displaystyle\int \frac 1 {\sqrt{a^2-x^2}} \space dx\)
\(\displaystyle \sin^{-1} {\frac x a} +C \ (a >0)\)
(11'') \(\displaystyle\int \frac 1 {\sqrt{1-a^2x^2}} \space dx\)
\(\displaystyle\frac {1} {a} \sin^{-1} {(ax)} +C \ (a > 0)\)
(11''') \(\displaystyle\int \frac {1} {\sqrt{a^2-(px+q)^2}} \space dx\)
\(\displaystyle \frac{1}{p} \sin^{-1} \frac{px+q}{a} + C \ (a > 0) \)
(12) \(\displaystyle\int \frac 1 {1+x^2} \space dx\)
\(\tan^{-1} x +C\)
(12') \(\displaystyle\int \frac 1 {a^2+x^2} \space dx\)
\(\displaystyle\frac {1} {a} \tan^{-1} {\frac x a} +C \ (a > 0) \)
(12'') \(\displaystyle\int \frac 1 {1+a^2x^2} \space dx\)
\(\displaystyle\frac {1} {a} tan^{-1} {(ax)} +C \ (a > 0)\)
(12''') \(\displaystyle\int \frac {1} {a^2+(px+q)^2} \space dx\)
\(\displaystyle \frac{1}{pa} \tan^{-1} \frac{px+q}{a} + C \) ( \(a>0\) )
置換積分で求めた式
(13) \(\displaystyle\int \tan x \space dx\)
\( - \log {\left | \cos x \right |} +C\)
(13') \(\displaystyle\int \tan (px+q) \space dx\)
\( \displaystyle - \frac 1 p \log {\left | \cos (px+q) \right |} +C\)
(14) \(\displaystyle\int \frac 1 {\tan x} \space dx\)
\( \log {\left | \sin x \right |} +C \)
(14') \(\displaystyle\int \frac 1 {\tan {(px+q)}} \space dx\)
\( \displaystyle \frac 1 p \log {\left | \sin {(px+q)} \right |} +C \)
(15) \(\displaystyle\int \frac 1 {\sqrt {x^2+1}} \space dx\)
\( \log {\left | x+ \sqrt {x^2+1} \right |} +C \)
(15') \(\displaystyle\int \frac 1 {\sqrt {(px+q)^2+a}} \space dx\)
\( \displaystyle \frac 1 p \log {\left | (px+q)+ \sqrt {(px+q)^2+a} \right |} +C \)
部分積分で求めた式
(16) \(\displaystyle\int \log {|x|} \space dx\)
\( \displaystyle x \log {|x|} -x +C \)
(16') \(\displaystyle\int \log {|px+q|} \space dx\)
\( \displaystyle \frac 1 p \left \{ (px+q) \log {|px+q|} -(px+q) \right \} +C \)
(17) \(\displaystyle\int \log_a {|x|} \space dx\)
\( \displaystyle \frac {x \log {|x|} -x } {\log a} +C \ (a>0, a \neq 1)\)
(17') \(\displaystyle\int \log_a {|px+q|} \space dx\)
\( \displaystyle \frac 1 {p \log a} \left \{ (px+q) \log {|px+q|} -(px+q) \right \} +C \ (a>0, a \neq 1)\)
(18) \(\displaystyle\int \sin^{-1} x \space dx\)
\( \displaystyle x \sin^{-1} x +\sqrt {1-x^2} + C \)
(18') \(\displaystyle\int \sin^{-1} {(px+q)} \space dx\)
\( \displaystyle \frac 1 p \left \{ (px+q) \sin^{-1} {(px+q)} +\sqrt {1-(px+q)^2} \right \} + C \)
(19) \(\displaystyle\int \tan^{-1} x \space dx\)
\( \displaystyle x \tan^{-1} x - \frac 1 2 \log {\left |1+x^2 \right |} +C \)
(19') \(\displaystyle\int \tan^{-1} {(px+q)} \space dx\)
\( \displaystyle \frac 1 p \left \{ (px+q) \tan^{-1} {(px+q)} - \frac 1 2 \log {\left |1+(px+q)^2 \right |} \right \} +C \)
(20) \(\displaystyle\int e^x \sin x \space dx\)
\( \displaystyle \frac {e^x} 2 (\sin x - \cos x) +C \)
(20') \(\displaystyle\int e^{px+q} \sin (lx+m) \space dx\)
\( \displaystyle \frac {e^{px+q}} {p^2+l^2} \left \{ p \sin (lx+m) - l \cos (lx+m) \right \} +C\)
（以下，\( (p, l) \neq (0, 0) \)）
(21) \(\displaystyle\int e^x \cos x \space dx\)
\( \displaystyle \frac {e^x} 2 (\sin x + \cos x) +C \)
(21') \(\displaystyle\int e^{px+q} \cos (lx+m) \space dx\)
\( \displaystyle \frac {e^{px+q}} {p^2+l^2} \left \{ l \sin (lx+m) + p \cos (lx+m) \right \} +C\)
(22) \(\displaystyle\int \sqrt {x^2+1} \space dx\)
\( \displaystyle \frac 1 2 \left (x \sqrt {x^2+1} + \log { \left |x+ \sqrt {x^2+1} \right |} \right ) +C\)
(22') \(\displaystyle\int \sqrt {(px+q)^2+a} \space dx\)
\( \displaystyle \frac 1 {2p} \left \{ (px+q) \sqrt {(px+q)^2+a} + a \log { \left |(px+q) + \sqrt {(px+q)^2+a} \right |} \right \} +C\)
(23) \(\displaystyle\int \sqrt {1-x^2} \space dx\)
\( \displaystyle \frac 1 2 \left (x \sqrt {1-x^2} + \sin^{-1} x \right ) +C\)
(23') \(\displaystyle\int \sqrt {a-(px+q)^2} \space dx\)
\( \displaystyle \frac 1 {2p} \left \{ (px+q) \sqrt {a-(px+q)^2} + a \sin^{-1} {\frac {px+q} {\sqrt a} } \right \} +C \ (a > 0)\)
有理関数の積分で求めた式
(24) \(\displaystyle\int \frac{1}{x^2-1} \space dx\)
\(\displaystyle \frac{1}{2} \log { \left | \frac{ x-1 }{x+1} \right |} + C \)
(24') \(\displaystyle\int \frac{1}{(px+q)^2 - a^2} \space dx\)
\(\displaystyle \frac{1}{2pa} \log { \left | \frac{ (px+q)-a}{(px+q)+a } \right |} + C \ (a>0) \)
(25) \(\displaystyle\int \frac{1}{\sin x} \space dx\)
\( \displaystyle \frac{1}{2} \log { \left | \frac{ 1- \cos x}{1+ \cos x } \right | } + C\)
\( = \displaystyle \log { \left | \tan {\frac x 2} \right | } + C \)
(25') \(\displaystyle\int \frac{1}{\sin (px+q)} \space dx\)
\( \displaystyle \frac{1}{2p} \log { \left | \frac{1- \cos (px+q) }{1+ \cos (px+q)} \right | } + C\)
\( = \displaystyle \frac{1}{p} \log { \left | \tan {\frac {px+q} {2}} \right | } + C \)
(26) \(\displaystyle\int \frac{1}{\cos x} \space dx\)
\( \displaystyle \frac{1}{2} \log { \left | \frac{1+ \sin x }{1- \sin x} \right | } + C\)
\( = \displaystyle \log { \left | \frac {1+ \sin x} {\cos x} \right | } + C \)
(26') \(\displaystyle\int \frac{1}{\cos (px+q)} \space dx\)
\( \displaystyle \frac{1}{2p} \log { \left | \frac{ 1+ \sin (px+q) }{ 1- \sin (px+q)} \right | } + C\)
\( = \displaystyle \frac{1}{p} \log { \left | \frac {1+ \sin (px+q)} {\cos (px+q)} \right | } + C \)
授業プリントでナンバリングされていない式
(A) \(\displaystyle\int \frac{1}{x^2-a^2} \space dx\)
\(\displaystyle \frac{1}{2a} \log { \left | \frac{x-a}{x+a} \right | } + C \ (a \neq 0) \)
3 置換積分の公式
\( \displaystyle \int f(px+q) \space dx = \frac {1} {p} F(px+q) +C \)
\( \displaystyle \int {\left \{ g(x) \right \}^n} \space g'(x) \space dx = \frac {1} {n+1} \left \{ g(x) \right \} ^{n+1} + C \space (n \neq -1) \)
\( \displaystyle \int \frac {g'(x)} {g(x)} dx = \log {\left | g(x) \right |} + C \)
4 有理関数の積分の公式
\( \displaystyle \int \frac 1 {x-a} dx = \log {\left | x-a \right | } + C \)
\( \displaystyle \int \frac 1 {(x-a)^n} dx = \frac 1 {-n+1} (x-a)^{-n+1} + C \space (以下，n \geq 2) \)
\( \displaystyle \int \frac x {x^2+b^2} dx = \frac 1 2 \log {\left | x^2+b^2 \right | } + C \)
\( \displaystyle \int \frac x {(x^2+b^2)^n} dx = \frac 1 {2(-n+1)} (x^2+b^2)^{-n+1} + C \)
\( I_1 = \displaystyle \int \frac 1 {x^2+b^2} dx = \frac 1 b \tan^{-1} \frac x b + C \space (以下，b \neq 0) \)
\( I_n = \displaystyle \int \frac 1 {(x^2+b^2)^n} dx = \frac 1 {b^2} \left \{ \frac 1 {2(n-1)} \cdot \frac x {(x^2+b^2)^{n-1}} + \frac {2n-3} {2n-2} \space I_{n-1} \right \} + C \)