(10)
\( \int -\frac {1}{\sqrt {a^2-x^2}} dx\)\( =cos^{-1} (\frac{x}{a})\)
(11)
\( \int \frac {1}{{a^2+x^2}} dx\)\( =\frac {1}{a} tan^{-1} (\frac{x}{a})\)
(12)
\( \int sec^2 x =\frac {1}{{cos^2 x}} dx = tan x \)
(13)※1
\( \int sinh\ x dx = cosh x \)
(14)※1
\( \int cosh\ x dx = sinh x \)
(15)※1
\( \int tanh\ x dx = log\ (e^x+e^{-1}) \)
(16)
\( \int \frac{1}{{x^2-a^2}} dx\)\( = \frac{1}{2} log \left| \frac{x-a}{x+a} \right| \cdots (a \neq b) \)
(17)
\( \int \sqrt{a^2-x^2} dx\)\(= \frac{1}{2} \left( x \sqrt{a^2-x^2} + a^2 sin^{-1} \frac{x}{a} \right) \)
\( \cdots (a>0) \)
(18)
\( \int \sqrt{a+x^2} dx\) \(=\frac{1}{2} \left( x \sqrt{a+x^2}+ a \ log \left| x+ \sqrt {a+x^2}
\right| \right)\) \( \cdots (a+x^2 ≥0, \ a \neq 0) \)
(ヒント:部分積分により導出できる)
【参照先】
(19)
\( \int \frac{1}{\sqrt{a+x^2}} dx\)\(= \ log \left| x+ \sqrt {a+x^2} \right|\)
\( \cdots (a+x^2 ≥0, \ a \neq 0) \)
(20)
\( \int \frac{f'(x)}{f(x)} dx\)\( = \log |f(x)| \)
(21)
\( \int {f(x)}{f'(x)} dx\)\( = \frac {1}{2} [f(x)]^2 \)
\( Ex: \ \int sin\ x\ cos\ x dx\) \(=\int sin\ x (sin\ x)' dx =\frac{1}{2} sin^2\ x \)
(22)
\( \int f'(x)g(x) dx\)\(=f(x)g(x)-\int f(x) g'(x) dx \)
(23)
【導出の参考先】
\( \int \frac{1}{cos^2x}dx=tan\ x \)
(24)
【導出の参考先】
\( \int \frac{1}{1+cos\ x} dx \)\(= \frac{1}{2} log \left(\frac{1+sin\ x} {1- sin\ x} \right) \)
…(\( x \ne \frac{1}{2} \pi , x \ne \frac{3}{2} \pi \) )
(25)
【導出の参考先】
\( \int \frac{1}{sin\ x} dx \)\(= \frac{1}{2} log \left( \frac{1-cos\ x}{1+cos\ x} \right) \)
…(\( x \ne 0 , x \ne \pi \) )
(26)
\( \int \frac{1}{tan\ x}dx\) \(= log |sin\ x|\)
…(\( x \ne 0 , x \ne \pi \) )
\(( \because \int \frac{1}{tan\ x}dx\) \(=\int \frac{cos\ x}{sin\ x}dx\) \(=\int \frac{(sin\ x)'}{sin\ x}dx \)
\(=log\ |sin\ x| )\)
(27)
【導出の参考先】
\(I_n=\int \frac{1}{(x^2+a^2)^n}dx\) \(\ (n≥2)\)
\(=\frac{1}{2(n-1)a^2}\)\( \left[\frac{x}{(x^2+a^2)^{n-1}} +(2n-3)I_{n-1}\right] \)