ja

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

人気のある微分積分 >

integral of 1/(cos^2(x)(1+tan(x)))

解

\int \frac{1}{cos ^{2} ( x ) ( 1 + tan ( x ) )} d x

解

ln tan^{2} (\frac{x}{2}) - 2 tan (\frac{x}{2}) - 1 - ln tan (\frac{x}{2}) + 1 - ln tan (\frac{x}{2}) - 1 + C

解答ステップ

\int \frac{1}{cos ^{2} ( x ) ( 1 + tan ( x ) )} d x

三角関数の公式を使用して書き換える

= \int \frac{1}{cos ( x ) ( cos ( x ) + sin ( x ) )} d x

u置換積分法を適用する

= \int \frac{2 ( u ^{2} + 1 )}{( - u ^{2} + 2 u + 1 ) ( - u ^{2} + 1 )} d u

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{u ^{2} + 1}{( - u ^{2} + 2 u + 1 ) ( - u ^{2} + 1 )} d u

以下の部分分数を得る:

\frac{u ^{2} + 1}{( - u ^{2} + 2 u + 1 ) ( - u ^{2} + 1 )} : \frac{u - 1}{u ^{2} - 2 u - 1} - \frac{1}{2 ( u + 1 )} - \frac{1}{2 ( u - 1 )}

= 2 \cdot \int \frac{u - 1}{u ^{2} - 2 u - 1} - \frac{1}{2 ( u + 1 )} - \frac{1}{2 ( u - 1 )} d u

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int \frac{u - 1}{u ^{2} - 2 u - 1} d u - \int \frac{1}{2 ( u + 1 )} d u - \int \frac{1}{2 ( u - 1 )} d u)

\int \frac{u - 1}{u ^{2} - 2 u - 1} d u = \frac{1}{2} ln u^{2} - 2 u - 1

\int \frac{1}{2 ( u + 1 )} d u = \frac{1}{2} ln ∣ u + 1 ∣

\int \frac{1}{2 ( u - 1 )} d u = \frac{1}{2} ln ∣ u - 1 ∣

= 2 (\frac{1}{2} ln u^{2} - 2 u - 1 - \frac{1}{2} ln ∣ u + 1 ∣ - \frac{1}{2} ln ∣ u - 1 ∣)

代用を戻す

u = tan (\frac{x}{2})

= 2 (\frac{1}{2} ln tan^{2} (\frac{x}{2}) - 2 tan (\frac{x}{2}) - 1 - \frac{1}{2} ln tan (\frac{x}{2}) + 1 - \frac{1}{2} ln tan (\frac{x}{2}) - 1)

簡素化

2 (\frac{1}{2} ln tan^{2} (\frac{x}{2}) - 2 tan (\frac{x}{2}) - 1 - \frac{1}{2} ln tan (\frac{x}{2}) + 1 - \frac{1}{2} ln tan (\frac{x}{2}) - 1) : ln tan^{2} (\frac{x}{2}) - 2 tan (\frac{x}{2}) - 1 - ln tan (\frac{x}{2}) + 1 - ln tan (\frac{x}{2}) - 1

= ln tan^{2} (\frac{x}{2}) - 2 tan (\frac{x}{2}) - 1 - ln tan (\frac{x}{2}) + 1 - ln tan (\frac{x}{2}) - 1

定数を解答に追加する

= ln tan^{2} (\frac{x}{2}) - 2 tan (\frac{x}{2}) - 1 - ln tan (\frac{x}{2}) + 1 - ln tan (\frac{x}{2}) - 1 + C

グラフ

人気の例

integral of (3x^2-7x)/(3x+2)

\int \frac{3 x ^{2} - 7 x}{3 x + 2} d x

integral of (e^{-1/(x^2)})/(x^3)

\int \frac{e ^{- \frac{1}{x ^{2}}}}{x ^{3}} d x

integral of (cos(sqrt(x+1)))/(sqrt(x+1))

\int \frac{cos ( x + 1 )}{x + 1} d x

integral of tan^3(x/2)

\int tan^{3} (\frac{x}{2}) d x

integral of x^2(6x^3-7)^2

\int x^{2} (6 x^{3} - 7)^{2} d x