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 (r^3)/(sqrt(16+r^2))

解

\int \frac{r ^{3}}{16 + r ^{2}} d r

解

\frac{1}{3} (16 + r^{2})^{\frac{3}{2}} - 16 16 + r^{2} + C

解答ステップ

\int \frac{r ^{3}}{16 + r ^{2}} d r

u置換積分法を適用する

= \int u^{2} - 16 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

= \int u^{2} d u - \int 16 d u

\int u^{2} d u = \frac{u ^{3}}{3}

\int 16 d u = 16 u

= \frac{u ^{3}}{3} - 16 u

代用を戻す

u = 16 + r^{2}

= \frac{( 16 + r ^{2} ) ^{3}}{3} - 16 16 + r^{2}

簡素化

\frac{( 16 + r ^{2} ) ^{3}}{3} - 16 16 + r^{2} : \frac{1}{3} (16 + r^{2})^{\frac{3}{2}} - 16 16 + r^{2}

= \frac{1}{3} (16 + r^{2})^{\frac{3}{2}} - 16 16 + r^{2}

定数を解答に追加する

= \frac{1}{3} (16 + r^{2})^{\frac{3}{2}} - 16 16 + r^{2} + C

グラフ

人気の例

integral from 0 to 6 of |x-4|

\int_{0}^{6} ∣ x - 4 ∣ d x

derivative of ln((x+6^{11}))

\frac{d}{d x} (ln ((x + 6)^{11}))

integral from 0 to 0.1 of xarctan(3x)

\int_{0}^{0.1} x arctan (3 x) d x

derivative of 200ln(x/(40))

\frac{d}{d x} (200 ln (\frac{x}{40}))

integral of (3t+1)(t-2)

\int (3 t + 1) (t - 2) d t