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 x^2(x^2+6)^{(-1)/2}

解

\int x^{2} (x^{2} + 6)^{\frac{- 1}{2}} d x

解

6 (- \frac{1}{2} ln \frac{1}{6 ^{\frac{1}{2}}} x + \frac{1}{6} (6 + x^{2}) + \frac{1}{12} x (6 + x^{2})^{\frac{1}{2}}) + C

解答ステップ

\int x^{2} (x^{2} + 6)^{\frac{- 1}{2}} d x

三角関数による置換を適用する

= \int 6 tan^{2} (u) sec (u) d u

定数を除く:

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

= 6 \cdot \int tan^{2} (u) sec (u) d u

三角関数の公式を使用して書き換える

= 6 \cdot \int (- 1 + sec^{2} (u)) sec (u) d u

拡張

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

= 6 \cdot \int - sec (u) + sec^{3} (u) 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

= 6 (- \int sec (u) d u + \int sec^{3} (u) d u)

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

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

= 6 (- ln ∣ tan (u) + sec (u) ∣ + \frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

代用を戻す

u = arctan (\frac{1}{6 ^{\frac{1}{2}}} x)

= 6 (- ln tan (arctan (\frac{1}{6 ^{\frac{1}{2}}} x)) + sec (arctan (\frac{1}{6 ^{\frac{1}{2}}} x)) + \frac{1}{2} sec (arctan (\frac{1}{6 ^{\frac{1}{2}}} x)) tan (arctan (\frac{1}{6 ^{\frac{1}{2}}} x)) + \frac{1}{2} ln tan (arctan (\frac{1}{6 ^{\frac{1}{2}}} x)) + sec (arctan (\frac{1}{6 ^{\frac{1}{2}}} x)))

簡素化

6 (- ln tan (arctan (\frac{1}{6 ^{\frac{1}{2}}} x)) + sec (arctan (\frac{1}{6 ^{\frac{1}{2}}} x)) + \frac{1}{2} sec (arctan (\frac{1}{6 ^{\frac{1}{2}}} x)) tan (arctan (\frac{1}{6 ^{\frac{1}{2}}} x)) + \frac{1}{2} ln tan (arctan (\frac{1}{6 ^{\frac{1}{2}}} x)) + sec (arctan (\frac{1}{6 ^{\frac{1}{2}}} x))) : 6 (- \frac{1}{2} ln \frac{1}{6 ^{\frac{1}{2}}} x + \frac{1}{6} (6 + x^{2}) + \frac{1}{12} x (6 + x^{2})^{\frac{1}{2}})

= 6 (- \frac{1}{2} ln \frac{1}{6 ^{\frac{1}{2}}} x + \frac{1}{6} (6 + x^{2}) + \frac{1}{12} x (6 + x^{2})^{\frac{1}{2}})

定数を解答に追加する

= 6 (- \frac{1}{2} ln \frac{1}{6 ^{\frac{1}{2}}} x + \frac{1}{6} (6 + x^{2}) + \frac{1}{12} x (6 + x^{2})^{\frac{1}{2}}) + C

グラフ

人気の例

integral of-x^2+7

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

integral of 1/(1+ax)

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

integral of 1/(3sqrt(2-3x))

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

integral of (3x+5)^2(3)

\int (3 x + 5)^{2} (3) d x

integral of 1/(sqrt(x^2-16x))

\int \frac{1}{x ^{2} - 16 x} d x