MEMO

物理・数学

三角関数の積分

t=\tan \frac{x}{2}と置くのが定石。

\int \frac{1}{\sin x}dx \\ \frac{dt}{dx}=\frac{1}{2\cos^2 \frac{x}{2}} \\ \begin{array}{ll} \int \frac{1}{\sin x}dx & = \int \frac{2\cos^2\frac{x}{2}}{2\sin \frac{x}{2}\cos \frac{x}{2}}dt \\ & = \int \frac{\cos \frac{x}{2}}{\sin \frac{x}{2}}dt \\ & = \int \frac{1}{t}dt \\ & = \ln|t|+C \\ & = \ln|tan \frac{x}{2}| + C \end{array}