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 sqrt(x^2-2x-3)

解

\int x^{2} - 2 x - 3 d x

解

4 (- ln (\frac{1}{2} x^{2} - 2 x - 3 + x - 1) + \frac{1}{8} (x - 1) x^{2} - 2 x - 3 + \frac{1}{2} ln (\frac{x ^{2} - 2 x - 3 + x - 1}{2})) + C

解答ステップ

\int x^{2} - 2 x - 3 d x

平方完成する

x^{2} - 2 x - 3 : (x - 1)^{2} - 4

= \int (x - 1)^{2} - 4 d x

u置換積分法を適用する

= \int u^{2} - 4 d u

三角関数による置換を適用する

= \int 4 tan^{2} (v) sec (v) d v

定数を除く:

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

= 4 \cdot \int tan^{2} (v) sec (v) d v

三角関数の公式を使用して書き換える

= 4 \cdot \int (- 1 + sec^{2} (v)) sec (v) d v

拡張

(- 1 + sec^{2} (v)) sec (v) : - sec (v) + sec^{3} (v)

= 4 \cdot \int - sec (v) + sec^{3} (v) d v

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

= 4 (- \int sec (v) d v + \int sec^{3} (v) d v)

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

\int sec^{3} (v) d v = \frac{1}{2} sec (v) tan (v) + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣

= 4 (- ln ∣ tan (v) + sec (v) ∣ + \frac{1}{2} sec (v) tan (v) + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

代用を戻す

= 4 (- ln tan (arcsec (\frac{1}{2} (x - 1))) + sec (arcsec (\frac{1}{2} (x - 1))) + \frac{1}{2} sec (arcsec (\frac{1}{2} (x - 1))) tan (arcsec (\frac{1}{2} (x - 1))) + \frac{1}{2} ln tan (arcsec (\frac{1}{2} (x - 1))) + sec (arcsec (\frac{1}{2} (x - 1))))

簡素化

4 (- ln tan (arcsec (\frac{1}{2} (x - 1))) + sec (arcsec (\frac{1}{2} (x - 1))) + \frac{1}{2} sec (arcsec (\frac{1}{2} (x - 1))) tan (arcsec (\frac{1}{2} (x - 1))) + \frac{1}{2} ln tan (arcsec (\frac{1}{2} (x - 1))) + sec (arcsec (\frac{1}{2} (x - 1)))) : 4 (- ln (\frac{1}{2} x^{2} - 2 x - 3 + x - 1) + \frac{1}{8} (x - 1) x^{2} - 2 x - 3 + \frac{1}{2} ln (\frac{x ^{2} - 2 x - 3 + x - 1}{2}))

= 4 (- ln (\frac{1}{2} x^{2} - 2 x - 3 + x - 1) + \frac{1}{8} (x - 1) x^{2} - 2 x - 3 + \frac{1}{2} ln (\frac{x ^{2} - 2 x - 3 + x - 1}{2}))

定数を解答に追加する

= 4 (- ln (\frac{1}{2} x^{2} - 2 x - 3 + x - 1) + \frac{1}{8} (x - 1) x^{2} - 2 x - 3 + \frac{1}{2} ln (\frac{x ^{2} - 2 x - 3 + x - 1}{2})) + C

グラフ

人気の例

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

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

integral of tan((3x)/7)

\int tan (\frac{3 x}{7}) d x

integral of ((1-sin^2(x)))/(cos(x))

\int \frac{( 1 - sin ^{2} ( x ) )}{cos ( x )} d x

integral of 2x^3e^{x^4}

\int 2 x^{3} e^{x^{4}} d x

integral of 1/(sqrt(x+1)+\sqrt[3]{x+1)}

\int \frac{1}{x + 1 + 3 x + 1} d x