【高校数学Ⅲ】5-1-3 いろいろな不定積分｜要点まとめ

\(x^2=(x+1)(x-1)+1\)より、
\(\displaystyle =\int\frac{(x+1)(x-1)+1}{x+1}dx\)
\(\displaystyle =\int\left(x-1+\frac{1}{x+1}\right)dx\)
\(\displaystyle =\frac{1}{2}x^2-x+\log|x+1|+C\ \ \)(\(C\)は積分定数)

\(\displaystyle \frac{2}{x^2-1}=-\frac{1}{x+1}+\frac{1}{x-1}\)より、
\(\displaystyle =\int\left(-\frac{1}{x+1}+\frac{1}{x-1}\right)dx\)
\(\displaystyle =-\log|x+1|+\log|x-1|+C\)
\(\displaystyle =\log\left|\frac{x-1}{x+1}\right|+C\ \ \)(\(C\)は積分定数)

\(\displaystyle =\int(1-\cos^2x)\sin xdx\)
\(\cos x=t\)とおくと、\(-\sin xdx=dt\)なので、
\(\displaystyle =\int(t^2+1)dt\)
\(\displaystyle =\frac{1}{3}t^3-t+C\)
\(\displaystyle =\frac{1}{3}\cos^3x-\cos x+C\ \ \)(\(C\)は積分定数)

\(\displaystyle \cos^2x=\frac{1+\cos2x}{2}\)より、
\(\displaystyle =\int\frac{1+\cos2x}{2}dx\)
\(\displaystyle =\frac{1}{2}x+\frac{1}{4}\sin2x+C\ \ \)(\(C\)は積分定数)

\(\displaystyle \sin\alpha\cos\beta=\frac{1}{2}\{\sin(\alpha+\beta)+\sin(\alpha-\beta)\}\)より、
\(\displaystyle =\frac{1}{2}\int\{\sin(3x+2x)+\sin(3x-2x)\}dx\)
\(\displaystyle =\frac{1}{2}\int(\sin5x+\sin x)dx\)
\(\displaystyle =-\frac{1}{10}\cos5x-\frac{1}{2}\cos x+C\ \ \)(\(C\)は積分定数)