tanx のべき乗の不定積分

積分数Ⅲ


\(\displaystyle \int \tan x \,dx= -\log |\cos x|+C\)

\(\displaystyle \int \tan^2 x \,dx= \tan x -x + C\)

\(\displaystyle \int \tan^3 x \,dx=\dfrac{\tan^2 x}{2}+\log|\cos x|+C\)

\(\displaystyle \int \tan^4 x \,dx= \dfrac{\tan^3 x}{3}-\tan x + x+C\)

\(C\) は積分定数


タンジェントの積分でよく使う公式

・\(\tan x\) と\(\cos x\) の関係式
\(1+\tan^2 x=\dfrac{1}{\cos^2 x}\)

・\(\tan x\) の微分
\((\tan x)^{\prime}=\left(\dfrac{\sin x}{\cos x}\right)^{\prime}=\dfrac{1}{\cos^2 x}\)

・積分の定石
\(\displaystyle\int f^{\,n} f^{\,\prime}\,dx=\dfrac{1}{n+1}f^{\,n+1}+C\)

\(\displaystyle \int \dfrac{f^{\,\prime}}{f}\,dx =\log |f|+C \)

以下、冒頭の結果を計算していきます。

(1) tanxの不定積分

\(\displaystyle \int \tan x\,dx\)
\(=\displaystyle \int \dfrac{\sin x}{\cos x}\,dx\)
\(=\displaystyle \int \dfrac{-(\cos x)^{\prime}}{\cos x}\,dx\)
\(=\displaystyle \color{red}{-\log |\cos x|+C}\)

(2) tan^2xの不定積分

\(\displaystyle \int \tan^2 x\,dx\)
\(=\displaystyle \int \left(\dfrac{1}{\cos^2 x}-1\right)\,dx\)
\(=\displaystyle \int \{\,(\tan x)^{\prime}-1\,\}\,dx\)
\(=\color{red}{\tan x -x +C}\)

(3) tan^3xの不定積分

\(\displaystyle \int \tan^3 x\,dx\)
\(=\displaystyle \int \tan x \cdot \tan^2 x \,dx\)
\(=\displaystyle \int \tan x \,\left(\dfrac{1}{\cos^2 x}-1 \right) \,dx\)
\(=\displaystyle \int \tan x \, \{(\tan x)^{\prime}-1\}\,dx\)
\(=\displaystyle \underbrace{\int \tan x \,(\tan x)^{\prime}\,dx}_{ff’\textbf{型の積分}}-\underbrace{\int\tan x\,dx}_{\textbf{(1)で求めた}}\)
\(=\color{red}{\dfrac{\tan^2 x}{2}+\log |\cos x| + C}\)

(4) tan^4xの不定積分

\(\displaystyle \int \tan^4 x\,dx\)
\(=\displaystyle \int \tan^2 x \cdot \tan^2 x \,dx\)
\(=\displaystyle \int \tan^2 x \,\left(\dfrac{1}{\cos^2 x}-1 \right) \,dx\)
\(=\displaystyle \int \tan^2 x \, \{(\tan x)^{\prime}-1\}\,dx\)
\(=\displaystyle \underbrace{\int \tan^2 x \,(\tan x)^{\prime}\,dx}_{f^2f’\textbf{型の積分}}-\underbrace{\int\tan^2 x\,dx}_{\textbf{(2)で求めた}}\)
\(=\color{red}{\dfrac{\tan^3 x}{3}-\tan x +x + C}\)