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)/(sqrt(25+x^2))

解

\int \frac{x ^{2}}{25 + x ^{2}} d x

解

25 (- \frac{1}{2} ln (\frac{x + 25 + x ^{2}}{5}) + \frac{1}{50} x 25 + x^{2}) + C

解答ステップ

\int \frac{x ^{2}}{25 + x ^{2}} d x

三角関数による置換を適用する

= \int 25 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

= 25 \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

= 25 \cdot \int (- 1 + sec^{2} (u)) sec (u) d u

拡張

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

= 25 \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

= 25 (- \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) ∣

= 25 (- 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}{5} x)

= 25 (- ln tan (arctan (\frac{1}{5} x)) + sec (arctan (\frac{1}{5} x)) + \frac{1}{2} sec (arctan (\frac{1}{5} x)) tan (arctan (\frac{1}{5} x)) + \frac{1}{2} ln tan (arctan (\frac{1}{5} x)) + sec (arctan (\frac{1}{5} x)))

簡素化

25 (- ln tan (arctan (\frac{1}{5} x)) + sec (arctan (\frac{1}{5} x)) + \frac{1}{2} sec (arctan (\frac{1}{5} x)) tan (arctan (\frac{1}{5} x)) + \frac{1}{2} ln tan (arctan (\frac{1}{5} x)) + sec (arctan (\frac{1}{5} x))) : 25 (- \frac{1}{2} ln (\frac{x + 25 + x ^{2}}{5}) + \frac{1}{50} x 25 + x^{2})

= 25 (- \frac{1}{2} ln (\frac{x + 25 + x ^{2}}{5}) + \frac{1}{50} x 25 + x^{2})

定数を解答に追加する

= 25 (- \frac{1}{2} ln (\frac{x + 25 + x ^{2}}{5}) + \frac{1}{50} x 25 + x^{2}) + C

グラフ

人気の例

derivative of 7x^4-2x^3+8x+5

\frac{d}{d x} (7 x^{4} - 2 x^{3} + 8 x + 5)

limit as x approaches 0 of e^{2x}-1

x \to 0 lim (e^{2 x} - 1)

integral of (x+1)e^{4x+1}

\int (x + 1) e^{4 x + 1} d x

y^{^{'}} + 2 y = \frac{6}{5}

derivative y=8(x+2)^{3/2}

d er i v a t i v e y = 8 (x + 2)^{\frac{3}{2}}